INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II:
HODGE-ARAKELOV-THEORETIC
EVALUATION
Shinichi
Mochizuki
December
2020
Abstract.
In
the
present
paper,
which
is
the
second
in
a
series
of
four
pa-
pers,
we
study
the
Kummer
theory
surrounding
the
Hodge-Arakelov-theoretic
eval-
uation
—
i.e.,
evaluation
in
the
style
of
the
scheme-theoretic
Hodge-Arakelov
theory
established
by
the
author
in
previous
papers
—
of
the
[reciprocal
of
the
l-
th
root
of
the]
theta
function
at
l
-torsion
points
[strictly
speaking,
shifted
by
a
suitable
2-torsion
point],
for
l
≥
5
a
prime
number.
In
the
first
paper
of
the
series,
we
studied
“miniature
models
of
conventional
scheme
theory”,
which
we
referred
to
as
Θ
±ell
NF-Hodge
theaters,
that
were
associated
to
certain
data,
called
initial
Θ-data,
that
includes
an
elliptic
curve
E
F
over
a
number
field
F
,
together
with
a
prime
num-
ber
l
≥
5.
The
underlying
Θ-Hodge
theaters
of
these
Θ
±ell
NF-Hodge
theaters
were
glued
to
one
another
by
means
of
“Θ-links”,
that
identify
the
[reciprocal
of
the
l-th
root
of
the]
theta
function
at
primes
of
bad
reduction
of
E
F
in
one
Θ
±ell
NF-Hodge
theater
with
[2l-th
roots
of]
the
q-parameter
at
primes
of
bad
reduction
of
E
F
in
an-
other
Θ
±ell
NF-Hodge
theater.
The
theory
developed
in
the
present
paper
allows
one
×
μ
to
construct
certain
new
versions
of
this
“Θ-link”.
One
such
new
version
is
the
Θ
gau
-
link,
which
is
similar
to
the
Θ-link,
but
involves
the
theta
values
at
l-torsion
points,
rather
than
the
theta
function
itself.
One
important
aspect
of
the
constructions
×
μ
that
underlie
the
Θ
gau
-link
is
the
study
of
multiradiality
properties,
i.e.,
properties
of
the
“arithmetic
holomorphic
structure”
—
or,
more
concretely,
the
ring/scheme
structure
—
arising
from
one
Θ
±ell
NF-Hodge
theater
that
may
be
formulated
in
such
a
way
as
to
make
sense
from
the
point
of
the
arithmetic
holomorphic
structure
of
another
Θ
±ell
NF-Hodge
theater
which
is
related
to
the
original
Θ
±ell
NF-Hodge
×
μ
theater
by
means
of
the
[non-scheme-theoretic!]
Θ
gau
-link.
For
instance,
certain
of
the
various
rigidity
properties
of
the
étale
theta
function
studied
in
an
earlier
paper
by
the
author
may
be
intepreted
as
multiradiality
properties
in
the
context
of
the
theory
of
the
present
series
of
papers.
Another
important
aspect
of
the
constructions
×
μ
that
underlie
the
Θ
gau
-link
is
the
study
of
“conjugate
synchronization”
via
the
-symmetry
of
a
Θ
±ell
NF-Hodge
theater.
Conjugate
synchronization
refers
to
a
F
±
l
certain
system
of
isomorphisms
—
which
are
free
of
any
conjugacy
indeterminacies!
—
between
copies
of
local
absolute
Galois
groups
at
the
various
l-torsion
points
at
which
the
theta
function
is
evaluated.
Conjugate
synchronization
plays
an
impor-
tant
role
in
the
Kummer
theory
surrounding
the
evaluation
of
the
theta
function
at
l-torsion
points
and
is
applied
in
the
study
of
coricity
properties
of
[i.e.,
the
study
of
×
μ
objects
left
invariant
by]
the
Θ
gau
-link.
Global
aspects
of
conjugate
synchronization
require
the
resolution,
via
results
obtained
in
the
first
paper
of
the
series,
of
certain
technicalities
involving
profinite
conjugates
of
tempered
cuspidal
inertia
groups.
Typeset
by
AMS-TEX
1
2
SHINICHI
MOCHIZUKI
Contents:
Introduction
§1.
Multiradial
Mono-theta
Environments
§2.
Galois-theoretic
Theta
Evaluation
§3.
Tempered
Gaussian
Frobenioids
§4.
Global
Gaussian
Frobenioids
Introduction
In
the
following
discussion,
we
shall
continue
to
use
the
notation
of
the
In-
troduction
to
the
first
paper
of
the
present
series
of
papers
[cf.
[IUTchI],
§I1].
In
particular,
we
assume
that
are
given
an
elliptic
curve
E
F
over
a
number
field
F
,
together
with
a
prime
number
l
≥
5.
In
the
present
paper,
which
forms
the
sec-
ond
paper
of
the
series,
we
study
the
Kummer
theory
surrounding
the
Hodge-
Arakelov-theoretic
evaluation
[cf.
Fig.
I.1
below]
—
i.e.,
evaluation
in
the
style
of
the
scheme-theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII]
—
of
the
reciprocal
of
the
l-th
root
of
the
theta
function
Θ
v
def
=
√
−1
·
1
q
v
2
(m+
12
)
2
−1
·
m∈Z
1
(−1)
n
·
q
v
2
(n+
12
)
2
n+
12
·
U
v
−
1
l
n∈Z
[cf.
[EtTh],
Proposition
1.4;
[IUTchI],
Example
3.2,
(ii)]
at
l-torsion
points
[strictly
speaking,
shifted
by
a
suitable
2-torsion
point]
in
the
context
of
the
theory
of
Θ
±ell
NF-Hodge
theaters
developed
in
[IUTchI].
Here,
relative
to
the
notation
of
[IUTchI],
§I1,
v
∈
V
bad
;
q
v
denotes
the
q-parameter
at
v
of
the
given
elliptic
curve
E
F
over
a
number
field
F;
U
v
denotes
the
standard
multiplicative
coordinate
on
the
Tate
curve
obtained
by
localizing
E
F
at
v.
Let
q
be
a
2l-th
root
of
q
v
.
v
Then
these
“theta
values
at
l-torsion
points”
will,
up
to
a
factor
given
by
a
2l-th
root
of
unity,
turn
out
to
be
of
the
form
[cf.
Remark
2.5.1,
(i)]
q
[Frobenius-like!]
Frobenioid-theoretic
theta
function
evalu-
⇓
ation
[Frobenius-like!]
Frobenioid-theoretic
theta
values
j
2
v
Kummer
.........
Kummer
.........
[étale-like!]
Galois-theoretic
étale
theta
function
evalu-
⇓
ation
[étale-like!]
Galois-theoretic
theta
values
Fig.
I.1:
The
Kummer
theory
surrounding
Hodge-Arakelov-theoretic
evaluation
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
3
def
—
where
j
∈
{0,
1,
.
.
.
,
l
=
(l
−
1)/2},
so
j
is
uniquely
determined
by
its
image
def
j
∈
|F
l
|
=
F
l
/{±1}
=
{0}
F
l
[cf.
the
notation
of
[IUTchI],
§I1].
In
order
to
understand
the
significance
of
Kummer
theory
in
the
context
of
Hodge-Arakelov-theoretic
evaluation,
it
is
important
to
recall
the
notions
of
“Frobenius-like”
and
“étale-like”
mathematical
structures
[cf.
the
discussion
of
[IUTchI],
§I1].
In
the
present
series
of
papers,
the
Frobenius-like
structures
consti-
tuted
by
[the
monoidal
portions
of]
Frobenioids
—
i.e.,
more
concretely,
by
various
monoids
—
play
the
important
role
of
allowing
one
to
construct
gluing
isomor-
phisms
such
as
the
Θ-link
which
lie
outside
the
framework
of
conventional
scheme/ring
theory
[cf.
the
discussion
of
[IUTchI],
§I2].
Such
gluing
isomor-
phisms
give
rise
to
Frobenius-pictures
[cf.
the
discussion
of
[IUTchI],
§I1].
On
the
other
hand,
the
étale-like
structures
constituted
by
various
Galois
and
arith-
metic
fundamental
groups
give
rise
to
the
canonical
splittings
of
such
Frobenius-
pictures
furnished
by
corresponding
étale-pictures
[cf.
the
discussion
of
[IUTchI],
§I1].
In
[IUTchIII],
absolute
anabelian
geometry
will
be
applied
to
these
Galois
and
arithmetic
fundamental
groups
to
obtain
descriptions
of
alien
arithmetic
holomorphic
structures,
i.e.,
arithmetic
holomorphic
structures
that
lie
on
the
opposite
side
of
a
Θ-link
from
a
given
arithmetic
holomorphic
structure
[cf.
the
discussion
of
[IUTchI],
§I3].
Thus,
in
light
of
the
equally
crucial
but
substantially
different
roles
played
by
Frobenius-like
and
étale-like
structures
in
the
present
series
of
papers,
it
is
of
crucial
importance
to
be
able
to
relate
corresponding
Frobenius-like
and
étale-like
versions
of
vari-
ous
objects
to
one
another.
This
is
the
role
played
by
Kummer
theory.
In
particular,
in
the
present
paper,
we
shall
study
in
detail
the
Kummer
theory
that
relates
Frobenius-like
and
étale-
like
versions
of
the
theta
function
and
its
theta
values
at
l-torsion
points
to
one
another
[cf.
Fig.
I.1].
One
important
notion
in
the
theory
of
the
present
paper
is
the
notion
of
mul-
tiradiality.
To
understand
this
notion,
let
us
recall
the
étale-picture
discussed
in
[IUTchI],
§I1
[cf.
[IUTchI],
Fig.
I1.6].
In
the
context
of
the
present
paper,
we
shall
be
especially
interested
in
the
étale-like
version
of
the
theta
function
and
its
±ell
theta
values
constructed
in
each
D-Θ
±ell
NF-Hodge
theater
(−)
HT
D-Θ
NF
;
thus,
one
can
think
of
the
étale-picture
under
consideration
as
consisting
of
the
diagram
given
in
Fig.
I.2
below.
As
discussed
earlier,
we
shall
ultimately
be
interested
in
applying
various
absolute
anabelian
reconstruction
algorithms
to
the
various
arith-
metic
fundamental
groups
that
[implicitly]
appear
in
such
étale-pictures
in
order
to
obtain
descriptions
of
alien
holomorphic
structures,
i.e.,
descriptions
of
objects
that
arise
on
one
“spoke”
[i.e.,
“arrow
emanating
from
the
core”]
that
make
sense
from
the
point
of
view
of
another
spoke.
In
this
context,
it
is
natural
to
classify
the
various
algorithms
applied
to
the
arithmetic
fundamental
groups
lying
in
a
given
spoke
as
follows
[cf.
Example
1.7]:
·
We
shall
refer
to
an
algorithm
as
coric
if
it
in
fact
only
depends
on
input
data
arising
from
the
mono-analytic
core
of
the
étale-picture,
i.e.,
the
data
that
is
common
to
all
spokes.
4
SHINICHI
MOCHIZUKI
·
We
shall
refer
to
an
algorithm
as
uniradial
if
it
expresses
the
objects
constructed
from
the
given
spoke
in
terms
that
only
make
sense
within
the
given
spoke.
·
We
shall
refer
to
an
algorithm
as
multiradial
if
it
expresses
the
objects
constructed
from
the
given
spoke
in
terms
of
corically
constructed
objects,
i.e.,
objects
that
make
sense
from
the
point
of
view
of
other
spokes.
Thus,
multiradial
algorithms
are
compatible
with
simultaneous
execution
at
multiple
spokes
[cf.
Example
1.7,
(v);
Remark
1.9.1],
while
uniradial
algorithms
may
only
be
consistently
executed
at
a
single
spoke.
Ultimately,
in
the
present
series
of
papers,
we
shall
be
interested
—
relative
to
the
goal
of
obtaining
“descriptions
of
alien
holomorphic
structures”
—
in
the
establishment
of
multiradial
algorithms
for
constructing
the
objects
of
interest,
e.g.,
[in
the
context
of
the
present
paper]
the
étale-like
versions
of
the
theta
functions
and
the
corresponding
theta
values
discussed
above.
Typically,
in
order
to
obtain
such
multiradial
algorithms,
i.e.,
algorithms
that
make
sense
from
the
point
of
view
of
other
spokes,
it
is
necessary
to
allow
for
some
sort
of
“indeterminacy”
in
the
descriptions
that
appear
in
the
algorithms
of
the
objects
constructed
from
the
given
spoke.
étale-like
version
of
j
2
Θ
v
,
{q
}
v
...
...
|
étale-like
version
of
j
2
Θ
v
,
{q
}
—
(−)
D
>
—
v
...
étale-like
version
of
j
2
Θ
v
,
{q
}
v
|
...
étale-like
version
of
j
2
Θ
v
,
{q
}
v
Fig.
I.2:
Étale-picture
of
étale-like
versions
of
theta
functions,
theta
values
Relative
to
the
analogy
between
the
inter-universal
Teichmüller
theory
of
the
present
series
of
papers
and
the
classical
theory
of
holomorphic
structures
on
Riemann
surfaces
[cf.
the
discussion
of
[IUTchI],
§I4],
one
may
think
of
coric
algorithms
as
corresponding
to
constructions
that
depend
only
on
the
underlying
real
analytic
structure
on
the
Riemann
surface.
Then
uniradial
algorithms
cor-
respond
to
constructions
that
depend,
in
an
essential
way,
on
the
holomorphic
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
5
structure
of
the
given
Riemann
surface,
while
multiradial
algorithms
correspond
to
constructions
of
holomorphic
objects
associated
to
the
Riemann
surface
which
are
expressed
[perhaps
by
allowing
for
certain
indeterminacies!]
solely
in
terms
of
the
underlying
real
analytic
structure
of
the
Riemann
surface
—
cf.
Fig.
I.3
below;
the
discussion
of
Remark
1.9.2.
Perhaps
the
most
fundamental
motivat-
ing
example
in
this
context
is
the
description
of
“alien
holomorphic
structures”
by
means
of
the
Teichmüller
deformations
reviewed
at
the
beginning
of
[IUTchI],
§I4,
relative
to
“unspecified/indeterminate”
deformation
data
[i.e.,
consisting
of
a
nonzero
square
differential
and
a
dilation
factor].
Indeed,
for
instance,
in
the
case
of
once-punctured
elliptic
curves,
by
applying
well-known
facts
concerning
Te-
ichmüller
mappings
[cf.,
e.g.,
[Lehto],
Chapter
V,
Theorem
6.3],
it
is
not
difficult
to
formulate
the
classical
result
that
“the
homotopy
class
of
every
orientation-preserving
homeomorphism
be-
tween
pointed
compact
Riemann
surfaces
of
genus
one
‘lifts’
to
a
unique
Teichmüller
mapping”
in
terms
of
the
“multiradial
formalism”
discussed
in
the
present
paper
[cf.
Example
1.7].
[We
leave
the
routine
details
to
the
reader.]
abstract
algorithms
inter-universal
Teichmüller
theory
classical
complex
Teichmüller
theory
uniradial
algorithms
arithmetic
holomorphic
structures
holomorphic
structures
multiradial
algorithms
arithmetic
holomorphic
structures
described
in
terms
of
underlying
mono-analytic
structures
holomorphic
structures
described
in
terms
of
underlying
real
analytic
structures
coric
algorithms
underlying
mono-analytic
structures
underlying
real
analytic
structures
Fig.
I.3:
Uniradiality,
Multiradiality,
and
Coricity
One
interesting
aspect
of
the
theory
of
the
present
series
of
papers
may
be
seen
in
the
set-theoretic
function
arising
from
the
theta
values
considered
above
j
→
q
j
2
v
—
a
function
that
is
reminiscent
of
the
Gaussian
distribution
(R
)
x
→
2
e
−x
on
the
real
line.
From
this
point
of
view,
the
passage
from
the
Frobenius-
picture
to
the
canonical
splittings
of
the
étale-picture
[cf.
the
discussion
of
[IUTchI],
6
SHINICHI
MOCHIZUKI
§I1],
i.e.,
in
effect,
the
computation
of
the
Θ-links
that
occur
in
the
Frobenius-
picture
by
means
of
the
various
multiradial
algorithms
that
will
be
established
in
the
present
series
of
papers,
may
be
thought
of
[cf.
the
diagram
of
Fig.
I.2!]
as
a
sort
of
global
arithmetic/Galois-theoretic
analogue
of
the
computation
of
the
classical
Gaussian
integral
∞
e
−x
dx
=
2
√
π
−∞
via
the
passage
from
cartesian
coordinates,
i.e.,
which
correspond
to
the
Frobenius-
picture,
to
polar
coordinates,
i.e.,
which
correspond
to
the
étale-picture
—
cf.
the
discussion
of
Remark
1.12.5.
One
way
to
understand
the
difference
between
coricity,
multiradiality,
and
±
uniradiality
at
a
purely
combinatorial
level
is
by
considering
the
F
l
-
and
F
l
-
symmetries
discussed
in
[IUTchI],
§I1
[cf.
the
discussion
of
Remark
4.7.4
of
the
present
paper].
Indeed,
at
a
purely
combinatorial
level,
the
F
l
-symmetry
may
be
thought
of
as
consisting
of
the
natural
action
of
F
on
the
set
of
labels
|F
l
|
=
l
{0}
F
l
[cf.
the
discussion
of
[IUTchI],
§I1].
Here,
the
label
0
corresponds
to
the
[mono-analytic]
core.
Thus,
the
corresponding
étale-picture
consists
of
various
copies
of
|F
l
|
glued
together
along
the
coric
label
0
[cf.
Fig.
I.4
below].
In
particular,
the
various
actions
of
copies
of
F
l
on
corresponding
copies
of
|F
l
|
are
“compatible
with
simultaneous
execution”
in
the
sense
that
they
commute
with
one
another.
That
is
to
say,
at
least
at
the
level
of
labels,
the
F
l
-symmetry
is
multiradial.
...
...
|
—
0
—
Fig.
I.4:
Étale-picture
of
F
l
-symmetries
±
±
±
±
...
...
↓↑
±
±
±
±
→
←
0
←
→
±
±
±
±
Fig.
I.5:
Étale-picture
of
F
±
l
-symmetries
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
7
In
a
similar
vein,
at
a
purely
combinatorial
level,
the
F
±
l
-symmetry
may
be
thought
±
of
as
consisting
of
the
natural
action
of
F
l
on
the
set
of
labels
F
l
[cf.
the
discussion
of
[IUTchI],
§I1].
Here
again,
the
label
0
corresponds
to
the
[mono-analytic]
core.
Thus,
the
corresponding
étale-picture
consists
of
various
copies
of
F
l
glued
together
along
the
coric
label
0
[cf.
Fig.
I.5
above].
In
particular,
the
various
actions
of
on
corresponding
copies
of
F
l
are
“incompatible
with
simultaneous
copies
of
F
±
l
execution”
in
the
sense
that
they
clearly
fail
to
commute
with
one
another.
That
is
to
say,
at
least
at
the
level
of
labels,
the
F
±
l
-symmetry
is
uniradial.
Since,
ultimately,
in
the
present
series
of
papers,
we
shall
be
interested
in
the
establishment
of
multiradial
algorithms,
“special
care”
will
be
necessary
in
order
to
obtain
multiradial
algorithms
for
constructing
objects
related
to
the
a
priori
uniradial
F
±
l
-symmetry
[cf.
the
discussion
of
Remark
4.7.3
of
the
present
paper;
[IUTchIII],
Remark
3.11.2,
(i),
(ii)].
The
multiradiality
of
such
algorithms
will
be
closely
related
to
the
fact
that
the
F
±
l
-symmetry
is
applied
to
relate
the
various
copies
of
local
units
modulo
torsion,
i.e.,
“O
×μ
”
[cf.
the
notation
of
[IUTchI],
§1]
at
various
labels
∈
F
l
that
lie
in
various
spokes
of
the
étale-picture
[cf.
the
discussion
of
Remark
4.7.3,
(ii)].
This
contrasts
with
the
way
in
which
the
a
pri-
ori
multiradial
F
l
-symmetry
will
be
applied,
namely
to
treat
various
“weighted
volumes”
corresponding
to
the
local
value
groups
and
global
realified
Frobenioids
at
various
labels
∈
F
l
that
lie
in
various
spokes
of
the
étale-picture
[cf.
the
dis-
cussion
of
Remark
4.7.3,
(iii)].
Relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
[cf.
[IUTchI],
§I4],
various
aspects
of
the
F
±
l
-symmetry
are
reminiscent
of
the
additive
monodromy
over
the
ordinary
locus
of
the
canonical
curves
that
occur
in
p-adic
Teichmüller
the-
ory;
in
a
similar
vein,
various
aspects
of
the
F
l
-symmetry
may
be
thought
of
as
corresponding
to
the
multiplicative
monodromy
at
the
supersingular
points
of
the
canonical
curves
that
occur
in
p-adic
Teichmüller
theory
—
cf.
the
discussion
of
Remark
4.11.4,
(iii);
Fig.
I.7
below.
Before
discussing
the
theory
of
multiradiality
in
the
context
of
the
theory
of
Hodge-Arakelov-theoretic
evaluation
theory
developed
in
the
present
paper,
we
pause
to
review
the
theory
of
mono-theta
environments
developed
in
[EtTh].
One
starts
with
a
Tate
curve
over
a
mixed-characteristic
nonarchimedean
local
field.
The
mono-theta
environment
associated
to
such
a
curve
is,
roughly
speak-
ing,
the
Kummer-theoretic
data
that
arises
by
extracting
N
-th
roots
of
the
theta
trivialization
of
the
ample
line
bundle
associated
to
the
origin
over
suitable
tem-
pered
coverings
of
the
curve
[cf.
[EtTh],
Definition
2.13,
(ii)].
Such
mono-theta
environments
may
be
constructed
purely
group-theoretically
from
the
[arithmetic]
tempered
fundamental
group
of
the
once-punctured
elliptic
curve
determined
by
the
given
Tate
curve
[cf.
[EtTh],
Corollary
2.18],
or,
alternatively,
purely
category-
theoretically
from
the
tempered
Frobenioid
determined
by
the
theory
of
line
bundles
and
divisors
over
tempered
coverings
of
the
Tate
curve
[cf.
[EtTh],
Theorem
5.10,
(iii)].
Indeed,
the
isomorphism
of
mono-theta
environments
between
the
mono-
theta
environments
arising
from
these
two
constructions
of
mono-theta
environ-
ments
—
i.e.,
from
tempered
fundamental
groups,
on
the
one
hand,
and
from
tem-
pered
Frobenioids,
on
the
other
[cf.
Proposition
1.2
of
the
present
paper]
—
may
be
thought
of
as
a
sort
of
Kummer
isomorphism
for
mono-theta
environments
[cf.
Proposition
3.4
of
the
present
paper,
as
well
as
[IUTchIII],
Proposition
2.1,
(iii)].
One
important
consequence
of
the
theory
of
[EtTh]
asserts
that
mono-theta
8
SHINICHI
MOCHIZUKI
environments
satisfy
the
following
three
rigidity
properties:
(a)
cyclotomic
rigidity,
(b)
discrete
rigidity,
and
(c)
constant
multiple
rigidity
—
cf.
the
Introduction
to
[EtTh].
Discrete
rigidity
assures
one
that
one
may
work
with
Z-translates
[where
we
write
Z
for
the
copy
of
“Z”
that
acts
as
a
group
of
covering
transformations
on
the
tempered
coverings
involved],
as
opposed
to
Z-translates
[i.e.,
where
Z
∼
=
Z
denotes
the
profinite
completion
of
Z],
of
the
theta
function,
i.e.,
one
need
not
contend
with
Z-powers
of
canonical
multiplicative
coordinates
[i.e.,
“U
”],
or
q-parameters
[cf.
Remark
3.6.5,
(iii);
[IUTchIII],
Remark
2.1.1,
(v)].
Although
we
will
certainly
“use”
this
discrete
rigidity
throughout
the
theory
of
the
present
series
of
papers,
this
property
of
mono-theta
environments
will
not
play
a
particularly
prominent
role
in
the
theory
of
the
present
series
of
papers.
The
Z-powers
of
“U
”
and
“q”
that
would
occur
if
one
does
not
have
discrete
rigidity
may
be
compared
to
the
PD-
formal
series
that
are
obtained,
a
priori,
if
one
attempts
to
construct
the
canonical
parameters
of
p-adic
Teichmüller
theory
via
formal
integration.
Indeed,
PD-formal
power
series
become
necessary
if
one
attempts
to
treat
such
canonical
parameters
as
objects
which
admit
arbitrary
O-powers,
where
O
denotes
the
completion
of
the
local
ring
to
which
the
canonical
parameter
belongs
[cf.
the
discussion
of
Remark
3.6.5,
(iii);
Fig.
I.6
below].
Constant
multiple
rigidity
plays
a
somewhat
more
central
role
in
the
present
series
of
papers,
in
particular
in
relation
to
the
theory
of
the
log-link,
which
we
shall
discuss
in
[IUTchIII]
[cf.
the
discussion
of
Remark
1.12.2
of
the
present
paper;
[IUTchIII],
Remark
1.2.3,
(i);
[IUTchIII],
Proposition
3.5,
(ii);
[IUTchIII],
Remark
3.11.2,
(iii)].
Constant
multiple
rigidity
asserts
that
the
multiplicative
monoid
O
F
×
·
Θ
N
v
v
—
which
we
shall
refer
to
as
the
theta
monoid
—
generated
by
the
reciprocal
of
the
l-th
root
of
the
theta
function
and
the
group
of
units
of
the
ring
of
inte-
gers
of
the
base
field
F
v
[cf.
the
notation
of
[IUTchI],
§I1]
admits
a
canonical
splitting,
up
to
2l-th
roots
of
unity,
that
arises
from
evaluation
at
the
[2-]torsion
point
corresponding
to
the
label
0
∈
F
l
[cf.
Corollary
1.12,
(ii);
Proposition
3.1,
(i);
Proposition
3.3,
(i)].
Put
another
way,
this
canonical
splitting
is
the
splitting
.
The
theta
monoid
of
determined,
up
to
2l-th
roots
of
unity,
by
Θ
v
∈
O
F
×
·
Θ
N
v
v
the
above
display,
as
well
as
the
associated
canonical
splitting,
may
be
constructed
algorithmically
from
the
mono-theta
environment
[cf.
Proposition
3.1,
(i)].
Rela-
tive
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory,
these
canonical
splittings
may
be
thought
of
as
corresponding
to
the
canonical
coordinates
of
p-adic
Teichmüller
theory,
i.e.,
more
precisely,
to
the
fact
that
such
canonical
coordinates
are
also
completely
determined
without
any
constant
multiple
indeterminacies
—
cf.
Fig.
I.6
below;
Remark
3.6.5,
(iii);
[IUTchIII],
Remark
3.12.4,
(i).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
Mono-theta-theoretic
rigidity
property
in
inter-universal
Teichmüller
theory
Corresponding
phenomenon
in
p-adic
Teichmüller
theory
mono-theta-theoretic
constant
multiple
rigidity
lack
of
constant
multiple
indeterminacy
of
canonical
coordinates
on
canonical
curves
mono-theta-theoretic
cyclotomic
rigidity
lack
of
Z
×
-power
indeterminacy
of
canonical
coordinates
on
canonical
curves,
Kodaira-Spencer
isomorphism
multiradiality
of
mono-theta-theoretic
constant
multiple,
cyclotomic
rigidity
Frobenius-invariant
nature
of
canonical
coordinates
mono-theta-theoretic
discrete
rigidity
formal
=
“non-PD-formal”
nature
of
canonical
coordinates
on
canonical
curves
9
Fig.
I.6:
Mono-theta-theoretic
rigidity
properties
in
inter-universal
Teichmüller
theory
and
corresponding
phenomena
in
p-adic
Teichmüller
theory
Cyclotomic
rigidity
consists
of
a
rigidity
isomorphism,
which
may
be
con-
structed
algorithmically
from
the
mono-theta
environment,
between
·
the
portion
of
the
mono-theta
environment
—
which
we
refer
to
as
the
exterior
cyclotome
—
that
arises
from
the
roots
of
unity
of
the
base
field
and
·
a
certain
copy
of
the
once-Tate-twisted
Galois
module
“
Z(1)”
—
which
we
refer
to
as
the
interior
cyclotome
—
that
appears
as
a
subquotient
of
the
geometric
tempered
fundamental
group
[cf.
Definition
1.1,
(ii);
Remark
1.1.1;
Proposition
1.3,
(i)].
This
rigidity
is
remark-
able
—
as
we
shall
see
in
our
discussion
below
of
the
corresponding
multiradiality
10
SHINICHI
MOCHIZUKI
property
—
in
that
unlike
the
“conventional”
construction
of
such
cyclotomic
rigid-
ity
isomorphisms
via
local
class
field
theory
[cf.
Proposition
1.3,
(ii)],
which
requires
one
to
use
the
entire
monoid
with
Galois
action
G
v
O
F
,
the
only
portion
of
v
the
monoid
O
F
that
appears
in
this
construction
is
the
portion
[i.e.,
the
“exterior
v
cyclotome”]
corresponding
to
the
torsion
subgroup
O
F
μ
v
⊆
O
F
[cf.
the
notation
v
of
[IUTchI],
§I1].
This
construction
depends,
in
an
essential
way,
on
the
com-
mutator
structure
of
theta
groups,
but
constitutes
a
somewhat
different
approach
to
utilizing
this
commutator
structure
from
the
“classical
approach”
involving
irre-
ducibility
of
representations
of
theta
groups
[cf.
Remark
3.6.5,
(ii);
the
Introduction
to
[EtTh]].
One
important
aspect
of
this
dependence
on
the
commutator
structure
of
the
theta
group
is
that
the
theory
of
cyclotomic
rigidity
yields
an
explanation
for
the
importance
of
the
special
role
played
by
the
first
power
of
[the
reciprocal
of
the
l-th
root
of
]
the
theta
function
in
the
present
series
of
papers
[cf.
Remark
3.6.4,
(iii),
(iv),
(v);
the
Introduction
to
[EtTh]].
Relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory,
mono-
theta-theoretic
cyclotomic
rigidity
may
be
thought
of
as
corresponding
either
to
the
fact
that
the
canonical
coordinates
of
p-adic
Teichmüller
theory
are
completely
determined
without
any
Z
×
-power
indeterminacies
or
[roughly
equivalently]
to
the
Kodaira-Spencer
isomorphism
of
the
canonical
indigenous
bundle
—
cf.
Fig.
I.6;
Remark
3.6.5,
(iii);
Remark
4.11.4,
(iii),
(b).
The
theta
monoid
O
F
×
·
Θ
N
v
v
discussed
above
admits
both
étale-like
and
Frobenius-like
[i.e.,
Frobenioid-theo-
retic]
versions,
which
may
be
related
to
one
another
via
a
Kummer
isomorphism
[cf.
Proposition
3.3,
(i)].
The
unit
portion,
together
with
its
natural
Galois
action,
of
the
Frobenioid-theoretic
version
of
the
theta
monoid
G
v
O
F
×
v
forms
the
portion
at
v
∈
V
bad
of
the
F
×
-prime-strip
“F
×
mod
”
that
is
preserved,
up
to
isomorphism,
by
the
Θ-link
[cf.
the
discussion
of
[IUTchI],
§I1;
[IUTchI],
Theorem
A,
(ii)].
In
the
theory
of
the
present
paper,
we
shall
introduce
modified
versions
of
the
Θ-link
of
[IUTchI]
[cf.
the
discussion
of
the
“Θ
×μ
-,
Θ
×μ
gau
-links”
below],
which,
unlike
the
Θ-link
of
[IUTchI],
only
preserve
[up
to
isomorphism]
the
F
×μ
-prime-strips
—
i.e.,
which
consist
of
the
data
G
v
O
F
×μ
=
O
F
×
/O
F
μ
v
v
v
[cf.
the
notation
of
[IUTchI],
§I1]
at
v
∈
V
bad
—
associated
to
the
F
×
-prime-
strips
preserved
[up
to
isomorphism]
by
the
Θ-link
of
[IUTchI].
Since
this
data
is
only
preserved
up
to
isomorphism,
it
follows
that
the
topological
group
“G
v
”
must
be
regarded
as
being
only
known
up
to
isomorphism,
while
the
monoid
O
F
×μ
must
be
v
regarded
as
being
only
known
up
to
[the
automorphisms
of
this
monoid
determined
by
the
natural
action
of]
Z
×
.
That
is
to
say,
one
must
regard
the
data
G
v
O
F
×μ
as
subject
to
Aut(G
v
)-,
Z
×
-indeterminacies.
v
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
11
These
indeterminacies
will
play
an
important
role
in
the
theory
of
the
present
series
of
papers
—
cf.
the
indeterminacies
“(Ind1)”,
“(Ind2)”
of
[IUTchIII],
Theorem
3.11,
(i).
Now
let
us
return
to
our
discussion
of
the
various
mono-theta-theoretic
rigidity
properties.
The
key
observation
concerning
these
rigidity
properties,
as
reviewed
above,
in
the
context
of
the
Aut(G
v
)-,
Z
×
-indeterminacies
just
discussed,
is
the
following:
the
canonical
splittings,
via
“evaluation
at
the
zero
section”,
of
the
theta
monoids,
together
with
the
construction
of
the
mono-theta-theoretic
cyclotomic
rigidity
isomorphism,
are
compatible
with,
in
the
sense
that
they
are
left
unchanged
by,
the
Aut(G
v
)-,
Z
×
-indeterminacies
dis-
cussed
above
—
cf.
Corollaries
1.10,
1.12;
Proposition
3.4,
(i).
Indeed,
this
observation
consti-
tutes
the
substantive
content
of
the
multiradiality
of
mono-theta-theoretic
con-
stant
multiple/cyclotomic
rigidity
[cf.
Fig.
I.6]
and
will
play
an
important
role
in
the
statements
and
proofs
of
the
main
results
of
the
present
series
of
papers
[cf.
[IUTchIII],
Theorem
2.2;
[IUTchIII],
Corollary
2.3;
[IUTchIII],
Theorem
3.11,
(iii),
(c);
Step
(ii)
of
the
proof
of
[IUTchIII],
Corollary
3.12].
At
a
technical
level,
this
“key
observation”
simply
amounts
to
the
observation
that
the
only
portion
of
the
monoid
O
F
×
that
is
relevant
to
the
construction
of
the
canonical
splittings
and
v
cyclotomic
rigidity
isomorphism
under
consideration
is
the
torsion
subgroup
O
F
μ
,
v
which
[by
definition!]
maps
to
the
identity
element
of
O
F
×μ
,
hence
is
immune
to
v
the
various
indeterminacies
under
consideration.
That
is
to
say,
the
multiradiality
of
mono-theta-theoretic
constant
multiple/cyclotomic
rigidity
may
be
regarded
as
an
essentially
formal
consequence
of
the
triviality
of
the
natural
homomorphism
O
F
μ
v
O
F
×μ
→
v
[cf.
Remark
1.10.2].
After
discussing,
in
§1,
the
multiradiality
theory
surrounding
the
various
rigid-
ity
properties
of
the
mono-theta
environment,
we
take
up
the
task,
in
§2
and
§3,
of
establishing
the
theory
of
Hodge-Arakelov-theoretic
evaluation,
i.e.,
of
passing
[for
v
∈
V
bad
]
O
F
×
·
Θ
N
v
v
j
2
O
F
×
·
{q
}
N
j=1,...,l
v
v
from
theta
monoids
as
discussed
above
[i.e.,
the
monoids
on
the
left-hand
side
of
the
above
display]
to
Gaussian
monoids
[i.e.,
the
monoids
on
the
right-hand
side
of
the
above
display]
by
means
of
the
operation
of
“evaluation”
at
l-torsion
points.
Just
as
in
the
case
of
theta
monoids,
Gaussian
monoids
admit
both
étale-like
ver-
sions,
which
constitute
the
main
topic
of
§2,
and
Frobenius-like
[i.e.,
Frobenioid-
theoretic]
versions,
which
constitute
the
main
topic
of
§3.
Moreover,
as
discussed
at
the
beginning
of
the
present
Introduction,
it
is
of
crucial
importance
in
the
theory
of
the
present
series
of
papers
to
be
able
to
relate
these
étale-like
and
Frobenius-like
versions
to
one
another
via
Kummer
theory.
One
important
observation
in
this
12
SHINICHI
MOCHIZUKI
context
—
which
we
shall
refer
to
as
the
“principle
of
Galois
evaluation”
—
is
the
following:
it
is
essentially
a
tautology
that
this
requirement
of
compatibility
with
Kummer
theory
forces
any
sort
of
“evaluation
operation”
to
arise
from
restriction
to
Galois
sections
of
the
[arithmetic]
tempered
fundamental
groups
involved
[i.e.,
Galois
sections
of
the
sort
that
arise
from
rational
points
such
as
l-torsion
points!]
—
cf.
the
discussion
of
Remarks
1.12.4,
3.6.2.
This
tautology
is
interesting
both
in
light
of
the
history
surrounding
the
Section
Conjecture
in
anabelian
geom-
etry
[cf.
[IUTchI],
§I5]
and
in
light
of
the
fact
that
the
theory
of
[SemiAnbd]
that
is
applied
to
prove
[IUTchI],
Theorem
B
—
a
result
which
plays
an
important
role
in
the
theory
of
§2
of
the
present
paper!
[cf.
the
discussion
below]
—
may
be
thought
of
as
a
sort
of
“Combinatorial
Section
Conjecture”.
At
this
point,
we
remark
that,
unlike
the
theory
of
theta
monoids
discussed
above,
the
theory
of
Gaussian
monoids
developed
in
the
present
paper
does
not,
by
itself,
admit
a
multiradial
formulation
[cf.
Remarks
2.9.1,
(iii);
3.4.1,
(ii);
3.7.1].
In
order
to
obtain
a
multiradial
formulation
of
the
theory
of
Gaussian
monoids
—
which
is,
in
some
sense,
the
ultimate
goal
of
the
present
series
of
papers!
—
it
will
be
necessary
to
combine
the
theory
of
the
present
paper
with
the
theory
of
the
log-link
developed
in
[IUTchIII].
This
will
allow
us
to
obtain
a
multiradial
formulation
of
the
theory
of
Gaussian
monoids
in
[IUTchIII],
Theorem
3.11.
One
important
aspect
of
the
theory
of
Hodge-Arakelov-theoretic
evaluation
is
the
notion
of
conjugate
synchronization.
Conjugate
synchronization
refers
to
a
collection
of
“symmetrizing
isomorphisms”
between
the
various
copies
of
the
local
absolute
Galois
group
G
v
associated
to
the
labels
∈
F
l
at
which
one
evaluates
the
theta
function
[cf.
Corollaries
3.5,
(i);
3.6,
(i);
4.5,
(iii);
4.6,
(iii)].
We
shall
also
use
the
term
“conjugate
synchronization”
to
refer
to
similar
collections
of
“sym-
metrizing
isomorphisms”
for
copies
of
various
objects
[such
as
the
monoid
O
F
]
v
closely
related
to
the
absolute
Galois
group
G
v
.
With
regard
to
the
collections
of
isomorphisms
between
copies
of
G
v
,
it
is
of
crucial
importance
that
these
isomor-
phisms
be
completely
well-defined,
i.e.,
without
any
conjugacy
indeterminacies!
Indeed,
if
one
allows
conjugacy
indeterminacies
[i.e.,
put
another
way,
if
one
allows
oneself
to
work
with
outer
isomorphisms,
as
opposed
to
isomorphisms],
then
one
must
sacrifice
either
·
the
distinct
nature
of
distinct
labels
∈
|F
l
|
—
which
is
necessary
in
j
2
order
to
keep
track
of
the
distinct
theta
values
“q
”
for
distinct
j
—
or
·
the
crucial
compatibility
of
étale-like
and
Frobenius-like
versions
of
the
symmetrizing
isomorphisms
with
Kummer
theory
—
cf.
the
discussion
of
Remark
3.8.3,
(ii);
[IUTchIII],
Remark
1.5.1;
Step
(vii)
of
the
proof
of
[IUTchIII],
Corollary
3.12.
In
this
context,
it
is
also
of
interest
to
observe
that
it
follows
from
certain
elementary
combinatorial
considerations
that
one
must
require
that
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
13
·
these
symmetrizing
isomorphisms
arise
from
a
group
action,
i.e.,
such
as
the
F
±
l
-symmetry
—
cf.
the
discussion
of
Remark
3.5.2.
Moreover,
since
it
will
be
of
crucial
impor-
tance
to
apply
these
symmetrizing
isomorphisms,
in
[IUTchIII],
§1
[cf.,
especially,
[IUTchIII],
Remark
1.3.2],
in
the
context
of
the
log-link
—
whose
definition
de-
pends
on
the
local
ring
structures
at
v
∈
V
bad
[cf.
the
discussion
of
[AbsTopIII],
§I3]
—
it
will
be
necessary
to
invoke
the
fact
that
·
the
symmetrizing
isomorphisms
at
v
∈
V
bad
arise
from
conjugation
op-
erations
within
a
certain
[arithmetic]
tempered
fundamental
group
—
namely,
the
tempered
fundamental
group
of
X
v
[cf.
the
notation
of
[IUTchI],
§I1]
—
that
contains
Π
v
as
an
open
subgroup
of
finite
index
—
cf.
the
discussion
of
Remark
3.8.3,
(ii).
Here,
we
note
that
these
“conjugation
operations”
related
to
the
F
±
l
-symmetry
may
be
applied
to
establish
conjugate
synchronization
precisely
because
they
arise
from
conjugation
by
elements
of
the
geometric
tempered
fundamental
group
[cf.
Remark
3.5.2,
(iii)].
The
significance
of
establishing
conjugate
synchronization
—
i.e.,
subject
to
the
various
requirements
discussed
above!
—
lies
in
the
fact
that
the
resulting
symmetrizing
isomorphisms
allow
one
to
construct
the
crucial
coric
F
×μ
-prime-strips
—
i.e.,
the
F
×μ
-prime-strips
that
are
preserved,
up
to
isomorphism,
by
the
modi-
fied
versions
of
the
Θ-link
of
[IUTchI]
[cf.
the
discussion
of
the
“Θ
×μ
-,
Θ
×μ
gau
-links”
below]
that
are
introduced
in
§4
of
the
present
paper
[cf.
Corollary
4.10,
(i),
(iv);
[IUTchIII],
Theorem
1.5,
(iii);
the
discussion
of
[IUTchIII],
Remark
1.5.1,
(i)].
In
§4,
the
theory
of
conjugate
synchronization
established
in
§3
[cf.
Corollaries
3.5,
(i);
3.6,
(i)]
is
extended
so
as
to
apply
to
arbitrary
v
∈
V,
i.e.,
not
just
v
∈
V
bad
[cf.
Corollaries
4.5,
(iii);
4.6,
(iii)].
In
particular,
in
order
to
work
with
the
theta
value
labels
∈
F
l
in
the
context
of
the
F
±
l
-symmetry,
i.e.,
which
involves
the
action
F
l
F
±
l
on
the
labels
∈
F
l
,
one
must
avail
oneself
of
the
global
portion
of
the
Θ
±ell
-Hodge
theaters
that
appear.
Indeed,
this
global
portion
allows
one
to
synchronize
the
a
priori
independent
indeterminacies
with
respect
to
the
action
of
{±1}
on
the
good
X
]
—
cf.
the
discussion
of
Remark
4.5.3,
various
X
v
[for
v
∈
V
bad
],
−
→
[for
v
∈
V
v
(iii).
On
the
other
hand,
the
copy
of
the
arithmetic
fundamental
group
of
X
K
that
constitutes
this
global
portion
of
the
Θ
±ell
-Hodge
theater
is
profinite,
i.e.,
it
does
not
admit
a
“globally
tempered
version”
whose
localization
at
v
∈
V
bad
is
naturally
isomorphic
to
the
corresponding
tempered
fundamental
group
at
v.
One
important
consequence
of
this
state
of
affairs
is
that
in
order
to
apply
the
global
±-synchronization
afforded
by
the
Θ
±ell
-
Hodge
theater
in
the
context
of
the
theory
of
Hodge-Arakelov-theoretic
evaluation
at
v
∈
V
bad
relative
to
labels
∈
F
l
that
correspond
to
conju-
gacy
classes
of
cuspidal
inertia
groups
of
tempered
fundamental
groups
at
14
SHINICHI
MOCHIZUKI
v
∈
V
bad
,
it
is
necessary
to
compute
the
profinite
conjugates
of
such
tempered
cuspidal
inertia
groups
—
cf.
the
discussion
of
[IUTchI],
Remark
4.5.1,
as
well
as
Remarks
2.5.2
and
4.5.3,
(iii),
of
the
present
paper,
for
more
details.
This
is
precisely
what
is
achieved
by
the
application
of
[IUTchI],
Theorem
B
[i.e.,
in
the
form
of
[IUTchI],
Corollary
2.5;
cf.
also
[IUTchI],
Remark
2.5.2]
in
§2
of
the
present
paper.
As
discussed
above,
the
theory
of
Hodge-Arakelov-theoretic
evaluation
devel-
oped
in
§1,
§2,
§3
is
strictly
local
[at
v
∈
V
bad
]
in
nature.
Thus,
in
§4,
we
discuss
the
essentially
routine
extensions
of
this
theory,
e.g.,
of
the
theory
of
Gaussian
monoids,
to
the
“remaining
portion”
of
the
Θ
±ell
-Hodge
theater,
i.e.,
to
v
∈
V
good
,
as
well
as
to
the
case
of
global
realified
Frobenioids
[cf.
Corollaries
4.5,
(iv),
(v);
4.6,
(iv),
(v)].
We
also
discuss
the
corresponding
complements,
involving
the
theory
of
[IUTchI],
§5,
for
ΘNF-Hodge
theaters
[cf.
Corollaries
4.7,
4.8].
This
leads
naturally
to
the
construction
of
modified
versions
of
the
Θ-link
of
[IUTchI]
[cf.
Corollary
4.10,
(iii)].
These
modified
versions
may
be
described
as
follows:
·
The
Θ
×μ
-link
is
essentially
the
same
as
the
Θ-link
of
[IUTchI],
Theorem
A,
except
that
F
-prime-strips
are
replaced
by
F
×μ
-prime-strips
[cf.
[IUTchI],
Fig.
I1.2]
—
i.e.,
roughly
speaking,
the
various
local
“O
×
”
are
replaced
by
“O
×μ
=
O
×
/O
μ
”.
×μ
-link,
except
that
the
·
The
Θ
×μ
gau
-link
is
essentially
the
same
as
the
Θ
×μ
theta
monoids
that
give
rise
to
the
Θ
-link
are
replaced,
via
composition
with
a
certain
isomorphism
that
arises
from
Hodge-Arakelov-theoretic
eval-
uation,
by
Gaussian
monoids
[cf.
the
above
discussion!]
—
i.e.,
roughly
j
2
speaking,
the
various
“Θ
v
”
at
v
∈
V
bad
are
replaced
by
“{q
}
j=1,...,l
”.
v
The
basic
properties
of
the
Θ
×μ
-,
Θ
×μ
gau
-links,
including
the
corresponding
Frobenius-
and
étale-pictures,
are
summarized
in
Theorems
A,
B
below
[cf.
Corollaries
4.10,
4.11
for
more
details].
Relative
to
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory,
the
passage
from
the
Θ
×μ
-link
to
the
Θ
×μ
gau
-link
via
Hodge-Arakelov-theoretic
evaluation
may
be
thought
of
as
corresponding
to
the
passage
MF
∇
-objects
Galois
representations
in
the
case
of
the
canonical
indigenous
bundles
that
occur
in
p-adic
Teichmüller
theory
—
cf.
the
discussion
of
Remark
4.11.4,
(ii),
(iii).
In
particular,
the
corre-
sponding
passage
from
the
Frobenius-picture
associated
to
the
Θ
×μ
-link
to
the
Frobenius-picture
associated
to
the
Θ
×μ
gau
-link
—
or,
more
properly,
relative
to
the
point
of
view
of
[IUTchIII]
[cf.
also
the
discussion
of
[IUTchI],
§I4],
from
the
log-theta-lattice
arising
from
the
Θ
×μ
-link
to
the
log-theta-lattice
arising
from
the
Θ
×μ
gau
-link
—
corresponds
[i.e..,
relative
to
the
analogy
with
p-adic
Teichmüller
the-
ory]
to
the
passage
from
thinking
of
canonical
liftings
as
being
determined
by
canonical
MF
∇
-objects
to
thinking
of
canonical
liftings
as
being
determined
by
canonical
Galois
representations
[cf.
Fig.
I.7
below].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
15
In
this
context,
it
is
of
interest
to
note
that
this
point
of
view
is
precisely
the
point
of
view
taken
in
the
absolute
anabelian
reconstruction
theory
developed
in
[CanLift],
§3
[cf.
Remark
4.11.4,
(iii),
(a)].
Finally,
we
observe
that
from
this
point
of
view,
the
important
theory
of
conjugate
synchronization
via
the
F
±
l
-
symmetry
may
be
thought
of
as
corresponding
to
the
theory
of
the
deformation
of
the
canonical
Galois
representation
from
“mod
p
n
”
to
“mod
p
n+1
”
[cf.
Fig.
I.7
below;
the
discussion
of
Remark
4.11.4,
(iii),
(d)].
Property
related
to
Hodge-Arakelov-theoretic
evaluation
in
inter-universal
Teichmüller
theory
Corresponding
phenomenon
in
p-adic
Teichmüller
theory
passage
from
Θ
×μ
-link
to
×μ
Θ
gau
-link
passage
from
canonicality
via
MF
∇
-objects
to
canonicality
via
crystalline
Galois
representations
F
±
l
-,
F
l
-
symmetries
ordinary,
supersingular
monodromy
of
canonical
Galois
representation
conjugate
synchronization
via
F
±
l
-symmetry
deformation
of
canonical
Galois
representation
from
“mod
p
n
”
to
“mod
p
n+1
”
Fig.
I.7:
Properties
related
to
Hodge-Arakelov-theoretic
evaluation
in
inter-universal
Teichmüller
theory
and
corresponding
phenomena
in
p-adic
Teichmüller
theory
Certain
aspects
of
the
various
constructions
discussed
above
are
summarized
in
the
following
two
results,
i.e.,
Theorems
A,
B,
which
are
abbreviated
versions
of
Corollaries
4.10,
4.11,
respectively.
On
the
other
hand,
many
important
aspects
—
such
as
multiradiality!
—
of
these
constructions
do
not
appear
explicitly
in
Theorems
A,
B.
The
main
reason
for
this
is
that
it
is
difficult
to
formulate
“final
results”
concerning
such
aspects
as
multiradiality
in
the
absence
of
the
framework
that
is
to
be
developed
in
[IUTchIII].
Theorem
A.
(Frobenius-pictures
of
Θ
±ell
NF-Hodge
Theaters)
Fix
a
col-
lection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
Θ
±ell
NF
Θ
±ell
NF
†
‡
±ell
3.1.
Let
HT
;
HT
be
Θ
NF-Hodge
theaters
[relative
to
the
±ell
given
initial
Θ-data]
—
cf.
[IUTchI],
Definition
6.13,
(i).
Write
†
HT
D-Θ
NF
,
16
SHINICHI
MOCHIZUKI
±ell
HT
D-Θ
NF
for
the
associated
D-Θ
±ell
NF-Hodge
theaters
—
cf.
[IUTchI],
Definition
6.13,
(ii).
Then:
‡
(i)
(Constant
Prime-Strips)
By
applying
the
symmetrizing
isomorphisms,
with
respect
to
the
F
±
l
-symmetry,
of
Corollary
4.6,
(iii),
to
the
data
of
the
un-
±ell
derlying
Θ
±ell
-Hodge
theater
of
†
HT
Θ
NF
that
is
labeled
by
t
∈
LabCusp
±
(
†
D
),
one
may
construct,
in
a
natural
fashion,
an
F
-prime-strip
∼
†
=
(
†
C
,
Prime(
†
C
)
→
V,
†
F
,
{
†
ρ
F
,v
}
v∈V
)
that
is
equipped
with
a
natural
identification
isomorphism
of
F
-prime-strips
∼
†
F
→
†
F
and
the
F
-prime-strip
†
F
mod
between
F
mod
of
[IUTchI],
Theorem
A,
(ii);
this
isomorphism
induces
a
natural
identification
isomorphism
of
D
-
∼
prime-strips
†
D
→
†
D
>
between
the
D
-prime-strip
†
D
associated
to
†
F
and
the
D
-prime-strip
†
D
>
of
[IUTchI],
Theorem
A,
(iii).
†
(ii)
(Theta
and
Gaussian
Prime-Strips)
By
applying
the
constructions
of
Corollary
4.6,
(iv),
(v),
to
the
underlying
Θ-bridge
and
Θ
±ell
-Hodge
theater
of
±ell
†
HT
Θ
NF
,
one
may
construct,
in
a
natural
fashion,
F
-prime-strips
∼
†
F
env
=
(
†
C
env
,
Prime(
†
C
env
)
→
V,
†
F
env
,
{
†
ρ
env,v
}
v∈V
)
†
F
gau
=
(
†
C
gau
,
Prime(
†
C
gau
)
→
V,
†
F
gau
,
{
†
ρ
gau,v
}
v∈V
)
∼
that
are
equipped
with
a
natural
identification
isomorphism
of
F
-prime-strips
∼
†
†
→
†
F
tht
between
F
env
and
the
F
-prime-strip
F
tht
of
[IUTchI],
Theorem
A,
(ii),
as
well
as
an
evaluation
isomorphism
†
F
env
†
F
env
∼
→
†
F
gau
of
F
-prime-strips.
‡
×μ
(iii)
(Θ
×μ
-
and
Θ
×μ
(respectively,
†
F
×μ
;
†
F
×μ
)
gau
-Links)
Write
F
env
gau
for
the
F
×μ
-prime-strip
associated
to
the
F
-prime-strip
‡
F
(respectively,
†
†
×μ
∼
‡
×μ
F
env
;
†
F
→
F
as
gau
).
We
shall
refer
to
the
full
poly-isomorphism
F
env
×μ
the
Θ
-link
±ell
±ell
Θ
×μ
†
HT
Θ
NF
−→
‡
HT
Θ
NF
±ell
±ell
[cf.
the
“Θ-link”
of
[IUTchI],
Theorem
A,
(ii)]
from
†
HT
Θ
NF
to
‡
HT
Θ
NF
,
∼
and
to
the
full
poly-isomorphism
†
F
×μ
→
‡
F
×μ
—
which
may
be
regarded
as
gau
∼
→
‡
F
×μ
by
composition
being
obtained
from
the
full
poly-isomorphism
†
F
×μ
env
with
the
inverse
of
the
evaluation
isomorphism
of
(ii)
—
as
the
Θ
×μ
gau
-link
†
from
†
HT
Θ
±ell
NF
HT
Θ
±ell
to
‡
HT
Θ
±ell
NF
NF
Θ
×μ
gau
−→
‡
±ell
HT
Θ
NF
.
(iv)
(Coric
F
×μ
-Prime-Strips)
The
definition
of
the
unit
portion
of
the
theta
and
Gaussian
monoids
that
appear
in
the
construction
of
the
F
-prime-
†
strips
†
F
env
,
F
gau
of
(ii)
gives
rise
to
natural
isomorphisms
†
×μ
F
∼
→
†
×μ
F
env
∼
→
†
×μ
F
gau
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
17
†
of
the
F
×μ
-prime-strips
associated
to
the
F
-prime-strips
†
F
,
†
F
env
,
F
gau
.
Moreover,
by
composing
these
natural
isomorphisms
with
the
poly-isomorphisms
induced
on
the
respective
F
×μ
-prime-strips
by
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
(iii),
one
obtains
a
poly-isomorphism
†
×μ
F
∼
‡
×μ
→
F
which
coincides
with
the
full
poly-isomorphism
between
these
two
F
×μ
-prime-
strips
—
that
is
to
say,
“
(−)
F
×μ
”
is
an
invariant
of
both
the
Θ
×μ
-
and
Θ
×μ
gau
-links.
Finally,
this
full
poly-isomorphism
induces
the
full
poly-isomorphism
†
∼
D
‡
→
D
between
the
associated
D
-prime-strips;
we
shall
refer
to
this
poly-isomorphism
as
±ell
±ell
the
D-Θ
±ell
NF-link
from
†
HT
D-Θ
NF
to
‡
HT
D-Θ
NF
.
±ell
(v)
(Frobenius-pictures)
Let
{
n
HT
Θ
NF
}
n∈Z
be
a
collection
of
distinct
Θ
±ell
NF-Hodge
theaters
indexed
by
the
integers.
Then
by
applying
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
(iii),
we
obtain
infinite
chains
Θ
×μ
(n−1)
HT
Θ
Θ
×μ
gau
(n−1)
HT
Θ
.
.
.
−→
.
.
.
−→
±ell
±ell
NF
Θ
×μ
n
HT
Θ
NF
Θ
×μ
gau
n
HT
Θ
−→
−→
±ell
±ell
NF
Θ
×μ
(n+1)
HT
Θ
NF
Θ
×μ
gau
(n+1)
HT
Θ
−→
−→
±ell
±ell
NF
Θ
×μ
NF
Θ
×μ
gau
−→
.
.
.
−→
.
.
.
±ell
NF-Hodge
theaters
—
cf.
Fig.
I.8
below,
in
the
case
of
Θ
×μ
-/Θ
×μ
gau
-linked
Θ
-link.
Either
of
these
infinite
chains
may
be
represented
symbolically
as
of
the
Θ
×μ
gau
an
oriented
graph
Γ
→
...
•
→
•
→
•
→
...
Θ
×μ
gau
Θ
×μ
—
i.e.,
where
the
arrows
correspond
to
either
the
“
−→
’s”
or
the
“
−→
’s”,
and
±ell
the
“•’s”
correspond
to
the
“
n
HT
Θ
NF
”.
This
oriented
graph
Γ
admits
a
natural
action
by
Z
—
i.e.,
a
translation
symmetry
—
but
it
does
not
admit
arbitrary
permutation
symmetries.
For
instance,
Γ
does
not
admit
an
automorphism
that
switches
two
adjacent
vertices,
but
leaves
the
remaining
vertices
fixed.
n
--
...
HT
Θ
±ell
NF
2
1
..
.
2
n
q
n
q
(l
)
v
(n+1)
--
v
.
..
n
q
v
HT
Θ
±ell
1
2
..
.
2
(n+1)
q
(n+1)
q
(l
)
v
→
NF
v
(n+1)
q
v
Fig.
I.8:
Frobenius-picture
associated
to
the
Θ
×μ
gau
-link
--
...
18
SHINICHI
MOCHIZUKI
Theorem
B.
(Étale-pictures
of
Base-Θ
±ell
NF-Hodge
Theaters)
Suppose
that
we
are
in
the
situation
of
Theorem
A,
(v).
(i)
Write
...
D
−→
n
±ell
HT
D-Θ
NF
D
−→
(n+1)
±ell
HT
D-Θ
NF
D
−→
...
—
where
n
∈
Z
—
for
the
infinite
chain
of
D-Θ
±ell
NF-linked
D-Θ
±ell
NF-
Hodge
theaters
[cf.
Theorem
A,
(iv),
(v)]
induced
by
either
of
the
infinite
chains
of
Theorem
A,
(v).
Then
this
infinite
chain
induces
a
chain
of
full
poly-
isomorphisms
∼
∼
∼
.
.
.
→
n
D
→
(n+1)
D
→
.
.
.
[cf.
Theorem
A,
(iv)].
That
is
to
say,
“
(−)
D
”
forms
a
constant
invariant
—
i.e.,
a
“mono-analytic
core”
[cf.
the
discussion
of
[IUTchI],
§I1]
—
of
the
above
infinite
chain.
(ii)
If
we
regard
each
of
the
D-Θ
±ell
NF-Hodge
theaters
of
the
chain
of
(i)
as
a
spoke
emanating
from
the
mono-analytic
core
“
(−)
D
”
discussed
in
(i),
then
we
obtain
a
diagram
—
i.e.,
an
étale-picture
of
D-Θ
±ell
NF-Hodge
theaters
—
as
in
Fig.
I.9
below
[cf.
the
situation
discussed
in
[IUTchI],
Theorem
A,
(iii)].
Thus,
each
spoke
may
be
thought
of
as
a
distinct
“arithmetic
holomorphic
struc-
ture”
on
the
mono-analytic
core.
Finally,
[cf.
the
situation
discussed
in
[IUTchI],
Theorem
A,
(iii)]
this
diagram
satisfies
the
important
property
of
admitting
arbi-
trary
permutation
symmetries
among
the
spokes
[i.e.,
the
labels
n
∈
Z
of
the
D-Θ
±ell
NF-Hodge
theaters].
(iii)
The
constructions
of
(i)
and
(ii)
are
compatible,
in
the
evident
sense,
with
the
constructions
of
[IUTchI],
Theorem
A,
(iii),
relative
to
the
natural
iden-
∼
tification
isomorphisms
(−)
D
→
(−)
D
>
[cf.
Theorem
A,
(i)].
n
HT
D-Θ
...
n
±ell
NF
...
|
±ell
HT
D-Θ
NF
—
(−)
—
D
|
...
n
n
±ell
HT
D-Θ
...
±ell
HT
D-Θ
NF
Fig.
I.9:
Étale-picture
of
D-Θ
±ell
NF-Hodge
theaters
NF
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
19
Acknowledgements:
The
research
discussed
in
the
present
paper
profited
enormously
from
the
gen-
erous
support
that
the
author
received
from
the
Research
Institute
for
Mathematical
Sciences,
a
Joint
Usage/Research
Center
located
in
Kyoto
University.
At
a
personal
level,
I
would
like
to
thank
Fumiharu
Kato,
Akio
Tamagawa,
Go
Yamashita,
Mo-
hamed
Saı̈di,
Yuichiro
Hoshi,
Ivan
Fesenko,
Fucheng
Tan,
Emmanuel
Lepage,
Arata
Minamide,
and
Wojciech
Porowski
for
many
stimulating
discussions
concerning
the
material
presented
in
this
paper.
Also,
I
feel
deeply
indebted
to
Go
Yamashita,
Mohamed
Saı̈di,
and
Yuichiro
Hoshi
for
their
meticulous
reading
of
and
numerous
comments
concerning
the
present
paper.
Finally,
I
would
like
to
express
my
deep
gratitude
to
Ivan
Fesenko
for
his
quite
substantial
efforts
to
disseminate
—
for
in-
stance,
in
the
form
of
a
survey
that
he
wrote
—
the
theory
discussed
in
the
present
series
of
papers.
Notations
and
Conventions:
We
shall
continue
to
use
the
“Notations
and
Conventions”
of
[IUTchI],
§0.
20
SHINICHI
MOCHIZUKI
Section
1:
Multiradial
Mono-theta
Environments
In
the
present
§1,
we
review
the
theory
of
mono-theta
environments
devel-
oped
in
[EtTh]
and
give
a
“multiradial”
interpretation
of
this
theory,
which
will
be
of
substantial
importance
in
the
present
series
of
papers.
Roughly
speaking,
in
the
language
of
[AbsTopIII],
§I3,
this
interpretation
consists
of
the
computation
of
which
portion
of
the
various
objects
constructed
from
the
“arithmetic
holomorphic
structures”
of
various
Θ
±ell
NF-Hodge
theaters
may
be
glued
together,
in
a
fashion
consistent
with
the
constructions
of
the
objects
of
interest,
via
a
“mono-analytic”
[i.e.,
“arithmetic
real
analytic”]
core.
Put
another
way,
this
computation
may
be
thought
of
as
the
computation
of
what
one
arithmetic
holomorphic
structure
looks
like
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
that
is
only
related
to
the
original
arithmetic
holomorphic
structure
via
the
mono-analytic
core.
In
fact,
this
sort
of
computation
forms
one
of
the
central
themes
of
the
present
series
of
papers.
Let
N
∈
N
≥1
be
a
positive
integer;
l
an
odd
prime
number;
k
an
MLF
of
odd
residue
characteristic
p
=
l
that
contains
a
primitive
4l-th
root
of
unity;
k
an
algebraic
closure
of
k;
X
k
a
hyperbolic
curve
of
type
(1,
(Z/lZ)
Θ
)
[cf.
[EtTh],
Definition
2.5,
(i)]
over
k
that
admits
a
stable
model
over
the
ring
of
integers
O
k
of
k;
X
k
→
C
k
the
k-core
determined
by
X
k
[cf.
the
discussion
at
the
beginning
of
[EtTh],
§2].
Write
Π
tp
X
k
def
tp
for
the
tempered
fundamental
group
of
X
k
;
G
k
=
Gal(k/k);
Δ
tp
X
=
Ker(Π
X
k
k
G
k
)
⊆
Π
tp
for
the
geometric
tempered
fundamental
group
of
X
.
We
shall
use
X
k
k
def
similar
notation
for
objects
associated
to
C
k
.
Definition
1.1.
Let
M
Θ
be
a
mod
N
mono-theta
environment
[cf.
[EtTh],
Definition
2.13,
(ii)]
which
is
isomorphic
to
the
mod
N
model
mono-theta
environment
determined
by
X
k
;
write
Π
M
Θ
for
the
underlying
topological
group
of
M
Θ
[cf.
[EtTh],
Definition
2.13,
(ii),
(a)].
Then:
(i)
There
exist
functorial
algorithms
M
Θ
→
Π
Y
(M
Θ
);
M
Θ
→
Δ
Y
(M
Θ
);
M
Θ
→
Π
X
(M
Θ
);
M
Θ
→
Δ
X
(M
Θ
);
M
Θ
→
G(M
Θ
);
M
Θ
→
(l
·
Δ
Θ
)(M
Θ
);
M
Θ
→
Δ
M
Θ
;
M
Θ
→
Π
μ
(M
Θ
)
for
constructing
from
M
Θ
a
quotient
Π
M
Θ
Π
Y
(M
Θ
)
[cf.
[EtTh],
Corollary
2.18,
(iii)];
a
topological
group
Π
X
(M
Θ
)
which
is
isomorphic
to
Π
tp
X
and
con-
k
tains
Π
Y
(M
Θ
)
as
a
normal
open
subgroup
[cf.
[EtTh],
Corollary
2.18,
(iii)];
a
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
21
quotient
Π
X
(M
Θ
)
G(M
Θ
)
corresponding
to
G
k
[cf.
[EtTh],
Corollary
2.18,
(i)],
which
may
also
be
thought
of
as
a
quotient
Π
M
Θ
Π
Y
(M
Θ
)
G(M
Θ
);
a
def
closed
normal
subgroup
Δ
M
Θ
=
Ker(Π
M
Θ
G(M
Θ
))
⊆
Π
M
Θ
;
a
closed
normal
def
subgroup
Δ
Y
(M
Θ
)
=
Ker(Π
Y
(M
Θ
)
G(M
Θ
))
⊆
Π
Y
(M
Θ
);
a
closed
normal
sub-
def
group
Δ
X
(M
Θ
)
=
Ker(Π
X
(M
Θ
)
G(M
Θ
))
⊆
Π
X
(M
Θ
)
corresponding
to
Δ
tp
X
[cf.
k
[EtTh],
Corollary
2.18,
(i)];
a
subquotient
(l
·
Δ
Θ
)(M
Θ
)
of
Π
Y
(M
Θ
)
which
admits
a
natural
Π
X
(M
Θ
)-action
[hence
also
a
Π
Y
(M
Θ
)-action,
as
well
as,
by
composition,
a
Π
M
Θ
-action]
relative
to
which
it
is
abstractly
isomorphic
to
Z(1)
[cf.
[EtTh],
Corol-
def
lary
2.18,
(i)];
a
closed
normal
subgroup
Π
μ
(M
Θ
)
=
Ker(Π
M
Θ
Π
Y
(M
Θ
))
⊆
Π
M
Θ
[cf.
[EtTh],
Corollary
2.19,
(i)]
which
admits
a
natural
Π
X
(M
Θ
)-action
[hence
also
a
Π
Y
(M
Θ
)-action,
as
well
as,
by
composition,
a
Π
M
Θ
-action]
relative
to
which
it
is
abstractly
isomorphic
to
(Z/N
Z)(1).
Also,
we
recall
that
the
structure
of
M
Θ
determines
a
lifting
of
the
natural
outer
action
of
def
(l
·
Z)(M
Θ
)
=
Π
X
(M
Θ
)/Π
Y
(M
Θ
)
∼
=
Δ
X
(M
Θ
)/Δ
Y
(M
Θ
)
on
Δ
Y
(M
Θ
)
to
an
outer
action
of
(l
·
Z)(M
Θ
)
on
Δ
M
Θ
[cf.
[EtTh],
Definition
2.13,
(i),
(ii),
and
the
preceding
discussion;
[EtTh],
Proposition
2.14,
(i)].
(ii)
We
shall
refer
to
(l
·
Δ
Θ
)(M
Θ
)
(respectively,
Π
μ
(M
Θ
))
as
the
interior
(respectively,
exterior)
cyclotome
associated
to
M
Θ
.
By
[EtTh],
Corollary
2.19,
(i),
there
is
a
functorial
algorithm
for
constructing
from
M
Θ
a
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)(M
Θ
)
⊗
(Z/N
Z)
→
Π
μ
(M
Θ
)
between
the
reductions
modulo
N
of
the
interior
and
exterior
cyclotomes
associated
to
M
Θ
.
Remark
1.1.1.
In
light
of
its
importance
in
the
present
series
of
papers,
we
pause
to
review
the
mono-theta-theoretic
cyclotomic
rigidity
isomorphism
of
Definition
1.1,
(ii),
in
more
detail,
as
follows.
(i)
First,
we
recall
from
[EtTh],
Proposition
2.4
[cf.
also
the
construction
of
the
covering
“Y
log
→
X
log
”
at
the
beginning
of
[EtTh],
§1],
that
the
topological
group
Π
X
(M
Θ
)
determines
topological
groups
Π
Y
(M
Θ
),
Π
X
(M
Θ
),
and
Π
C
(M
Θ
)
—
i.e.,
corresponding
to
the
coverings
“Y
log
→
X
log
→
C
log
”
of
the
discussion
preceding
[EtTh],
Definition
2.7
—
all
of
which
[together
with
Π
X
(M
Θ
)]
may
be
regarded
as
open
subgroups
of
Π
C
(M
Θ
)
Π
Y
(M
Θ
)
⊆
Π
X
(M
Θ
)
⊆
Π
C
(M
Θ
)
(⊇
Π
X
(M
Θ
)
⊇
Π
X
(M
Θ
))
that
are
equipped
with
compatible
surjections
to
G(M
Θ
).
Write
Δ
Y
(M
Θ
)
⊆
Δ
X
(M
Θ
)
⊆
Δ
C
(M
Θ
)
(⊇
Δ
X
(M
Θ
)
⊇
Δ
X
(M
Θ
))
for
the
respective
kernels
of
these
surjections.
Moreover,
the
various
topological
groups
of
the
above
two
displays
are
equipped
with
subquotients
denoted
by
means
22
SHINICHI
MOCHIZUKI
of
a
superscript
“Θ”
or
a
superscript
“ell”
[cf.
the
discussion
at
the
beginning
of
[EtTh],
§1].
These
subquotients
are
completely
determined
by
the
topological
group
structure
of
Π
C
(M
Θ
)
[cf.
the
discussion
at
the
beginning
of
[EtTh],
§1;
the
proof
of
[EtTh],
Proposition
1.8].
For
instance,
we
observe
that
one
may
reconstruct
from
the
topological
group
Π
X
(M
Θ
)
[cf.
[EtTh],
Corollary
2.18,
(i)]
the
quotient
Θ
Π
M
Θ
Π
Y
(M
Θ
)
Π
ell
Y
(M
)
[which
isomorphic
to
Z(1)
G
k
,
relative
to
the
natural
cyclotomic
action
of
G
k
tp
ell
on
Z(1)]
corresponding
to
the
quotient
“Π
tp
Y
(Π
Y
)
”
of
the
discussion
at
the
beginning
of
[EtTh],
§1.
(ii)
Observe
that
any
closed
subgroup
H
⊆
Π
Y
(M
Θ
)
determines,
by
forming
the
inverse
image
via
the
quotient
Π
M
Θ
Π
Y
(M
Θ
),
a
closed
subgroup
Π
M
Θ
|
H
⊆
Π
M
Θ
.
On
the
other
hand,
by
forming
the
quotient
of
Π
M
Θ
by
the
restriction
of
the
“theta
section
portion”
of
the
mono-theta
environment
M
Θ
[cf.
[EtTh],
Definition
2.13,
Θ
Θ
(ii),
(c)]
to
the
subgroup
Ker(Π
Y
(M
Θ
)
Π
Θ
Y
(M
))
⊆
Π
Y
(M
),
it
makes
sense
to
speak
of
the
quotient
of
Π
M
Θ
(Π
M
Θ
)
Π
M
Θ
|
Π
Θ
(M
Θ
)
Y
Θ
(
Π
Θ
Y
(M
))
Θ
determined
by
the
quotient
Π
Y
(M
Θ
)
Π
Θ
Y
(M
)
—
cf.
the
discussion
at
the
beginning
of
the
proof
of
[EtTh],
Corollary
2.19,
(i).
In
particular,
it
makes
sense
to
speak
of
the
subquotient
of
Π
M
Θ
determined
by
any
closed
subgroup
—
i.e.,
such
Θ
Θ
Θ
as
(l
·
Δ
Θ
)(M
Θ
)
⊆
Π
Θ
Y
(M
)
—
of
Π
Y
(M
).
(iii)
In
addition
to
the
subgroup
Π
μ
(M
Θ
)
→
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
determined
by
the
subgroup
Π
μ
(M
Θ
)
⊆
Π
M
Θ
of
Definition
1.1,
(i),
the
“theta
section
portion”
of
the
mono-theta
environment
M
Θ
[cf.
[EtTh],
Definition
2.13,
(ii),
(c)]
determines,
by
restriction,
a
subgroup
s
Θ
(M
Θ
)|
(l·Δ
Θ
)(M
Θ
)
⊆
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
that
maps
isomorphically
to
(l
·Δ
Θ
)(M
Θ
)
via
the
natural
projection
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
(l
·
Δ
Θ
)(M
Θ
)
[cf.
the
proof
of
[EtTh],
Corollary
2.19,
(i)].
On
the
other
hand,
by
considering
liftings
γ
of
automorphisms
of
Δ
Y
(M
Θ
)
determined
by
conjugation
by
elements
of
Δ
X
(M
Θ
)
to
automorphisms
of
Π
M
Θ
that
determine
outer
automor-
phisms
of
the
sort
that
appear
in
the
definition
of
a
mono-theta
environment
[cf.
[EtTh],
Definition
2.13,
(ii),
(b)]
and
then
forming
the
“commutator
γ(β)
·
β
−1
”
of
such
liftings
with
arbitrary
elements
β
∈
Δ
Y
(M
Θ
)
[cf.
[EtTh],
Proposition
2.14,
(i)],
one
obtains
a
natural
bilinear
“commutator
map”
Θ
[−,
−]
:
(Δ
X
(M
Θ
)/Δ
Y
(M
Θ
))
×
Δ
ell
Y
(M
)
→
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
∼
—
where
we
recall
that
(l
·
Z)
→
Δ
X
(M
Θ
)/Δ
Y
(M
Θ
)
is
abstractly
isomorphic
to
Z,
Θ
while
Δ
ell
Y
(M
)
is
abstractly
isomorphic
to
Z
—
whose
image
determines
a
subgroup
s
alg
(M
Θ
)|
(l·Δ
Θ
)(M
Θ
)
⊆
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
23
that
maps
isomorphically
to
(l
·Δ
Θ
)(M
Θ
)
via
the
natural
projection
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
(l
·
Δ
Θ
)(M
Θ
)
[cf.
the
proof
of
[EtTh],
Corollary
2.19,
(i)].
The
mono-theta-
theoretic
cyclotomic
rigidity
isomorphism
of
Definition
1.1,
(ii),
is
then
re-
constructed
[cf.
[EtTh],
Corollary
2.19,
(i)]
by
forming
the
difference
of
the
two
sections
s
Θ
(M
Θ
)|
(l·Δ
Θ
)(M
Θ
)
,
s
alg
(M
Θ
)|
(l·Δ
Θ
)(M
Θ
)
.
(iv)
Next,
we
observe
that
the
mono-theta-theoretic
cyclotomic
rigidity
isomor-
phism
of
Definition
1.1,
(ii),
admits
a
certain
symmetry
with
respect
to
the
group
Δ
C
(M
Θ
)/Δ
X
(M
Θ
)
∼
[cf.
[IUTchI],
Definition
6.1,
(v)],
as
follows.
First
of
all,
=
F
±
l
let
us
observe
that
the
natural
conjugation
action
of
Π
Y
(M
Θ
)
on
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
factors
through
the
natural
surjection
Π
Y
(M
Θ
)
G(M
Θ
).
In
particular,
by
ap-
plying
the
natural
surjection
Π
C
(M
Θ
)
G(M
Θ
),
one
may
regard
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
as
being
equipped
with
a
“naively
defined”
action
by
Π
C
(M
Θ
).
On
the
other
hand,
let
us
recall
from
the
discussion
preceding
[EtTh],
Definition
2.13,
that
the
“model”
for
Π
M
Θ
is
originally
constructed
as
the
subgroup
Π
μ
(M
Θ
)
Π
Y
(M
Θ
)
⊆
Π
μ
(M
Θ
)
Π
C
(M
Θ
)
—
where
the
semi-direct
products
are
formed
relative
to
the
natural
cyclotomic
action
of
Π
C
(M
Θ
).
Here,
the
evident
subquotient
Π
μ
(M
Θ
)
(l
·
Δ
Θ
)(M
Θ
)
of
Π
μ
(M
Θ
)
Π
C
(M
Θ
)
—
i.e.,
which
corresponds
to
the
subquotient
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
of
Π
M
Θ
—
is
easily
verified
to
be
stabilized
by
the
action
via
conjugation
of
Π
μ
(M
Θ
)
Π
C
(M
Θ
).
Moreover,
one
verifies
easily
that
this
conjugation
action
of
Π
μ
(M
Θ
)
Π
C
(M
Θ
)
factors
through
the
natural
quotient
Π
μ
(M
Θ
)
Π
C
(M
Θ
)
Π
C
(M
Θ
)
G(M
Θ
)
and
coincides
with
the
action
of
G(M
Θ
)
via
the
cyclotomic
character
G(M
Θ
)
→
Z
×
on
the
abelian
profinite
group
Π
μ
(M
Θ
)
(l
·
Δ
Θ
)(M
Θ
)
[where
we
re-
call
that
Z
×
acts
tautologically
on
any
abelian
profinite
group].
That
is
to
say,
in
summary,
even
if
one
is
not
equipped
with
the
“model
embedding”
Π
M
Θ
→
Π
μ
(M
Θ
)
Π
C
(M
Θ
),
the
“naively
defined”
action
of
Π
C
(M
Θ
)
on
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
is
in
fact
a
“natural
action”
in
the
sense
that
it
necessarily
coincides
with
the
natural
conjugation
action
arising
from
this
“model
embedding”.
Next,
let
us
observe
that
the
inclusion
Δ
X
(M
Θ
)
⊆
Δ
X
(M
Θ
)
induces
natural
isomorphisms
∼
Δ
X
(M
Θ
)/Δ
Y
(M
Θ
)
→
Δ
X
(M
Θ
)/Δ
Y
(M
Θ
),
∼
Θ
ell
Θ
Δ
ell
Y
(M
)
→
Δ
Y
(M
)
of
subquotients
of
Π
C
(M
Θ
),
whose
codomains
are
[unlike
the
domains
of
these
isomorphisms!]
stabilized
by
the
conjugation
action
of
Π
C
(M
Θ
).
In
particular,
by
applying
these
natural
isomorphisms,
one
may
regard
the
“commutator
map”
of
(iii)
as
a
map
Θ
[−,
−]
:
(Δ
X
(M
Θ
)/Δ
Y
(M
Θ
))
×
Δ
ell
Y
(M
)
→
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
—
i.e.,
a
map
for
which
both
the
domain
and
the
codomain
are
equipped
with
natural
actions
by
Π
C
(M
Θ
).
Now
one
verifies
easily
that
this
“commutator
map”
is
equivariant
with
respect
to
these
natural
actions
by
Π
C
(M
Θ
),
and,
24
SHINICHI
MOCHIZUKI
moreover,
that
the
various
subgroups
of
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
constructed
in
(iii)
are
stabilized
by
the
natural
action
by
Π
C
(M
Θ
).
In
this
context,
it
is
also
of
in-
terest
to
note
that,
in
fact,
it
follows
immediately
from
a
similar
argument
to
the
argument
concerning
the
automorphisms
of
a
mono-theta
environment
given
in
the
proof
of
[EtTh],
Corollary
2.18,
(iv),
that
up
to
composition
with
auto-
morphisms
of
Π
M
Θ
that
differ
from
the
identity
automorphism
by
a
twisted
homo-
Θ
Θ
morphism
Π
M
Θ
Π
Y
(M
Θ
)
Π
ell
Y
(M
)
→
Π
μ
(M
)
that
arises
from
a
Kummer
l
class
of
a
product
of
integral
powers
of
“(
Ü
)
2
”
and
“q
X
2
”
[cf.
[EtTh],
Proposi-
tion
1.4,
(ii)]
—
i.e.,
automorphisms
that
have
no
effect
on
the
construction
of
the
“commutator
map”
of
the
above
display!
—
the
“model
embedding”
Π
M
Θ
→
Π
μ
(M
Θ
)
Π
C
(M
Θ
)
may
be
reconstructed
algorithmically
from
the
mono-theta
environment
M
Θ
.
Thus,
in
summary,
the
various
constructions
discussed
in
(iii)
that
underlie
the
mono-theta-
theoretic
cyclotomic
rigidity
isomorphism
of
Definition
1.1,
(ii),
are
stabilized
by
the
natural
action
by
Π
C
(M
Θ
),
hence,
in
particular,
by
the
natural
action
by
(Π
C
(M
Θ
)
⊇)
Δ
C
(M
Θ
)
Δ
C
(M
Θ
)/Δ
X
(M
Θ
)
∼
=
F
±
l
.
Here,
we
remark
that
the
fact
that
these
constructions
are
stabilized
by
the
ac-
tion
of
Δ
X
(M
Θ
)
is
“less
interesting”
in
the
sense
that
the
automorphisms
of
Π
X
(M
Θ
)
that
arise
from
the
conjugation
action
by
Δ
X
(M
Θ
)
lift
[indeed,
“almost
uniquely”!
—
cf.
[EtTh],
Corollary
2.18,
(iv)]
to
automorphisms
of
M
Θ
,
hence
stabilize
the
constructions
under
consideration
as
a
consequence
of
the
functoriality
of
these
constructions
with
respect
to
automorphisms
[cf.
[EtTh],
Corollary
2.19,
(i)].
It
is
for
this
reason
that,
in
the
present
context,
it
is
natural
to
regard
the
symmetry
properties
of
interest
as
being
symmetries
with
respect
to
the
quotient
Δ
C
(M
Θ
)
Δ
C
(M
Θ
)/Δ
X
(M
Θ
)
∼
=
F
±
l
.
On
the
other
hand,
the
approach
of
the
above
discussion
via
model
embeddings
to
this
full
symmetry
with
respect
to
F
±
l
may
also
be
regarded
as
being
simply
an
explicit
computation,
in
the
case
of
this
F
±
l
-symmetry,
of
the
functoriality
of
the
constructions
under
consideration
with
respect
to
isomorphisms
[cf.
[EtTh],
Corollary
2.19,
(i)].
(v)
In
the
context
of
the
discussion
following
the
final
display
of
(iv),
it
is
perhaps
of
interest
to
recall
that
the
symmetries
of
mono-theta
environments
relative
to
the
conjugation
action
by
Δ
X
(M
Θ
)
are
a
consequence
of
the
“shift-
ing
automorphisms”
discussed
in
[EtTh],
Proposition
2.14,
(ii)
[cf.
the
discussion
of
[EtTh],
Remark
2.14.3].
That
is
to
say,
despite
the
fact
that
the
meromor-
phic
function
constituted
by
the
theta
function
does
not
admit
such
symmetries,
the
corresponding
mono-theta
environment
does
admit
such
symmetries.
This
is
one
important
difference
between
the
theory
of
mono-theta
environments
and
the
theory
of
bi-theta
environments
[cf.
the
discussion
of
[EtTh],
Remark
2.14.3].
Alternatively,
the
existence
of
such
symmetries
may
be
regarded
as
one
of
the
fundamental
differences
between
the
mono-theta-theoretic
approach
to
cyclotomic
rigidity
taken
in
[EtTh]
and
the
approach
to
cyclotomic
rigidity
taken
in
[IUTchI],
Example
5.1,
(v),
via
Kummer
classes
of
rational
functions.
Put
another
way,
this
fundamental
difference
may
be
thought
of
as
the
difference
between
constructing
a
cyclotomic
rigidity
isomorphism
from
a
line
bundle
—
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
25
i.e.,
which,
in
general,
admits
more
symmetries
than
a
rational
function
—
and
constructing
a
cyclotomic
rigidity
isomorphism
from
a
rational
function.
On
the
other
hand,
if
one
attempts
to
mimick
the
approach
of
[EtTh]
[i.e.,
of
constructing
“shifting
automorphisms”
as
in
[EtTh],
Proposition
2.14,
(ii)]
in
the
case
of
sym-
metries
with
respect
to
the
quotient
Δ
C
(M
Θ
)
Δ
C
(M
Θ
)/Δ
X
(M
Θ
)
∼
=
F
±
l
,
then
it
is
necessary
to
allow
“denominators
of
the
form
1
l
”
when
one
works
with
the
module
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
.
In
fact,
however,
when
one
computes
the
commutator
map
[−,
−]
considered
in
(iv),
such
terms
with
denominators
vanish,
as
a
consequence
of
the
fact
that
Π
M
Θ
|
(l·Δ
Θ
)(M
Θ
)
commutes
with
the
elements
of
interest
in
the
com-
putation
of
this
commutator
map.
It
is
precisely
this
state
of
affairs
that
allows
one
to
construct
an
F
±
l
-symmetric
cyclotomic
rigidity
isomorphism
as
dis-
cussed
in
(iv),
that
is
to
say,
which,
by
itself,
is
somewhat
weaker
than
the
“full
mono-theta
environment”
[i.e.,
which
does
not
admit
F
±
l
-symmetries
unless
one
allows
for
denominators
as
discussed
above!].
Thus,
in
summary,
by
comparison
to
the
approach
to
cyclotomic
rigidity
taken
in
[EtTh],
the
slightly
weaker
approach
discussed
in
(iv)
may
be
thought
of
as
corresponding
to
the
difference
between
con-
structing
a
cyclotomic
rigidity
isomorphism
from
a
line
bundle
and
constructing
a
cyclotomic
rigidity
isomorphism
from
the
curvature,
or
first
Chern
class,
of
the
line
bundle
[cf.
the
discussion
of
Remark
3.6.5
below].
One
key
property
of
mono-theta
environments
is
that
they
may
be
constructed
either
group-theoretically
from
Π
tp
X
or
category-theoretically
from
certain
tempered
Frobenioids
related
to
X
k
.
k
Proposition
1.2.
(Group-
and
Frobenioid-theoretic
Constructions
of
Mono-theta
Environments)
(i)
Let
Π
be
a
topological
group
isomorphic
to
Π
tp
X
.
Then
there
exists
a
functorial
group-theoretic
algorithm
k
Π
→
M
Θ
(Π)
for
constructing
from
the
topological
group
Π
a
mod
N
mono-theta
environ-
ment
“up
to
isomorphism”
[cf.
[EtTh],
Corollary
2.18,
(ii)]
such
that
the
composite
of
this
algorithm
with
the
algorithm
M
Θ
(Π)
→
Π
X
(M
Θ
(Π))
discussed
in
∼
Definition
1.1,
(i),
admits
a
functorial
isomorphism
Π
→
Π
X
(M
Θ
(Π)).
Here,
the
“isomorphism
indeterminacy”
of
M
Θ
(Π)
is
with
respect
to
a
group
of
“μ
N
-
conjugacy
classes”
of
automorphisms
which
is
of
order
1
(respectively,
2)
if
N
is
odd
(respectively,
even)
[cf.
[EtTh],
Corollary
2.18,
(iv)].
(ii)
Let
C
be
a
category
equivalent
to
the
tempered
Frobenioid
determined
by
X
k
[i.e.,
the
Frobenioid
denoted
“C”
in
the
discussion
at
the
beginning
of
[EtTh],
§5;
the
Frobenioid
denoted
“F
v
”
in
the
discussion
of
[IUTchI],
Example
3.2,
(i)].
Thus,
C
admits
a
natural
Frobenioid
structure
over
a
base
category
D
equivalent
0
to
B
temp
(Π
tp
X
)
[cf.
[FrdI],
Corollary
4.11,
(ii),
(iv);
[EtTh],
Proposition
5.1].
k
Then
there
exists
a
functorial
algorithm
C
→
M
Θ
(C)
26
SHINICHI
MOCHIZUKI
for
constructing
from
the
category
C
a
mod
N
mono-theta
environment
[cf.
[EtTh],
Theorem
5.10,
(iii)]
such
that
the
composite
of
this
algorithm
with
the
algo-
rithm
M
Θ
(C)
→
Π
X
(M
Θ
(C))
discussed
in
Definition
1.1,
(i),
admits
a
functorial
∼
isomorphism
D
→
B
temp
(Π
X
(M
Θ
(C)))
0
.
Proof.
The
assertions
of
Proposition
1.2
follow
immediately
from
the
results
of
[EtTh]
that
are
quoted
in
the
statements
of
these
assertions.
The
cyclotomic
rigidity
isomorphism
of
Definition
1.1,
(ii),
that
arises
in
the
case
of
the
mono-theta
environment
M
Θ
(C)
constructed
from
the
tempered
Frobe-
nioid
C
[cf.
Proposition
1.2,
(ii)]
is
compatible
with
a
certain
cyclotomic
rigidity
isomorphism
that
arises
in
the
theory
of
[AbsTopIII]
[cf.
also
[FrdII],
Theorem
2.4,
(ii)]
in
the
following
sense.
Proposition
1.3.
(Compatibility
of
Cyclotomic
Rigidity
Isomorphisms)
In
the
situation
of
Proposition
1.2,
(ii):
(i)
(Mono-theta
Environments
Associated
to
Tempered
Frobenioids)
For
a
suitable
object
S
∈
Ob(C)
[cf.
[EtTh],
Lemma
5.9,
(v)],
whose
image
in
D
we
denote
by
S
bs
∈
Ob(D),
the
interior
cyclotome
(l
·
Δ
Θ
)(M
Θ
(C))
⊗
(Z/N
Z)
corresponds
to
a
certain
subquotient
of
Aut(S
bs
),
which
we
denote
by
(l
·
Δ
Θ
)
S
⊗
(Z/N
Z),
while
the
exterior
cyclotome
Π
μ
(M
Θ
(C))
corresponds
to
the
subgroup
μ
N
(S)
⊆
O
×
(S)
⊆
Aut(S).
In
particular,
the
cyclotomic
rigidity
isomorphism
of
Definition
1.1,
(ii),
takes
the
form
of
an
isomorphism
∼
(l
·
Δ
Θ
)
S
⊗
(Z/N
Z)
→
μ
N
(S)
(∗
mono-Θ
)
[cf.
[EtTh],
Proposition
5.5;
[EtTh],
Lemma
5.9,
(v)].
(ii)
(MLF-Galois
Pairs)
Relative
to
the
formal
correspondence
between
p-
adic
Frobenioids
[such
as
the
base-field-theoretic
hull
C
bs-fld
associated
to
C
—
cf.
[EtTh],
Definition
3.6,
(iv)]
and
“MLF-Galois
TM-pairs”
in
the
theory
of
[AbsTopIII]
[cf.
[AbsTopIII],
Remark
3.1.1],
“μ
N
(S)”
[cf.
(i)]
corresponds
to
“μ
Z
(M
TM
)
⊗
(Z/N
Z)”
in
the
theory
of
[AbsTopIII],
§3
[cf.
[AbsTopIII],
Definition
3.1,
(v)],
while
“(l
·
Δ
Θ
)
S
⊗
(Z/N
Z)”
[cf.
(i)]
corresponds
to
“μ
Z
(Π
X
)
⊗
(Z/N
Z)”
in
the
theory
of
[AbsTopIII],
§1
[cf.
[AbsTopIII],
Theorem
1.9,
(b);
[AbsTopIII],
Remark
1.10.1,
(ii);
[IUTchI],
Remark
3.1.2,
(iii)].
In
particular,
by
composing
the
∼
inverse
of
the
natural
isomorphism
“μ
Z
(G
k
)
→
μ
Z
(Π
X
)”
of
[AbsTopIII],
Corollary
∼
1.10,
(c),
with
the
inverse
of
the
natural
isomorphism
“μ
Z
(M
TM
)
→
μ
Z
(G)”
of
[Ab-
sTopIII],
Remark
3.2.1,
we
obtain
another
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)
S
⊗
(Z/N
Z)
→
μ
N
(S)
(∗
bs-Gal
)
[cf.
the
various
identifications/correspondences
of
notation
discussed
above].
(iii)
(Compatibility)
The
cyclotomic
rigidity
isomorphisms
(∗
mono-Θ
),
(∗
bs-Gal
)
of
[EtTh],
[AbsTopIII]
[cf.
(i),
(ii)]
coincide.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
27
Proof.
Assertions
(i),
(ii)
follow
immediately
from
the
results
and
definitions
of
[EtTh],
[AbsTopIII]
that
are
quoted
in
the
statements
of
these
assertions.
Assertion
(iii)
follows
immediately
from
the
fact
that
in
the
situation
where
the
Frobenioid
C
involved
is
not
just
“some
abstract
category”,
but
rather
arises
from
familiar
ob-
jects
of
scheme
theory
[cf.
the
theory
of
[EtTh],
§1!],
both
isomorphisms
(∗
mono-Θ
),
(∗
bs-Gal
)
coincide
with
the
conventional
identification
between
the
cyclotomes
in-
volved
that
arises
from
conventional
scheme
theory.
Proposition
1.4.
(Étale
Theta
Functions
of
Standard
Type)
Let
Π
be
as
in
Proposition
1.2,
(i).
Then
there
are
functorial
group-theoretic
algorithms
[cf.
[EtTh],
Corollary
2.18,
(i)]
Π
→
Π
Ÿ
(Π);
Π
→
(l
·
Δ
Θ
)(Π)
for
constructing
from
Π
the
open
subgroup
Π
Ÿ
(Π)
⊆
Π
corresponding
to
the
tem-
pered
covering
“
Ÿ
”
[cf.
the
discussion
preceding
[EtTh],
Definition
2.7]
and
a
cer-
tain
subquotient
(l
·
Δ
Θ
)(Π)
of
Π
[cf.
the
subquotient
“(l
·
Δ
Θ
)(M
Θ
)”
of
Definition
1.1,
(i)],
as
well
as
a
functorial
group-theoretic
algorithm
Π
→
θ(Π)
⊆
H
1
(Π
Ÿ
(Π),
(l
·
Δ
Θ
)(Π))
—
cf.
the
constant
multiple
rigidity
property
of
[EtTh],
Corollary
2.19,
(iii)
—
for
constructing
from
Π
the
set
θ(Π)
of
μ
l
-multiples
[i.e.,
where
μ
l
denotes
the
group
of
l-th
roots
of
unity]
of
the
reciprocal
of
the
“(l
·
Z
×
μ
2
)-orbit
η̈
Θ,l·Z×μ
2
of
an
l-th
root
of
the
étale
theta
function
of
standard
type”
of
[EtTh],
Definition
2.7.
In
this
context,
we
shall
write
∞
θ(Π)
⊆
1
lim
−→
J
H
(Π
Ÿ
(Π)|
J
,
(l
·
Δ
Θ
)(Π))
—
where
∞
θ(Π)
denotes
the
subset
of
elements
of
the
direct
limit
of
cohomology
modules
in
the
display
for
which
some
[positive
integer]
multiple
[i.e.,
some
[pos-
itive
integer]
power,
if
one
writes
these
modules
“multiplicatively”]
coincides,
up
to
torsion,
with
an
element
of
θ(Π);
J
ranges
over
the
finite
index
open
subgroups
of
Π;
the
notation
“|
J
”
denotes
the
fiber
product
“×
Π
J”.
Proof.
The
assertions
of
Proposition
1.4
follow
immediately
from
the
results
and
definitions
of
[EtTh]
that
are
quoted
in
the
statements
of
these
assertions.
Remark
1.4.1.
Before
proceeding,
let
us
recall
from
[EtTh],
§1,
§2,
the
theory
surrounding
the
“étale
theta
functions
of
standard
type”
that
appeared
in
Proposi-
tion
1.4.
(i)
Write
X
k
→
X
k
→
C
k
for
the
hyperbolic
orbicurves
of
type
(1,
l-tors),
(1,
l-tors)
±
determined
by
X
k
[cf.
[EtTh],
Proposition
2.4].
Thus,
X
k
has
a
unique
zero
cusp
[i.e.,
the
unique
cusp
fixed
by
the
action
of
the
Galois
group
Gal(X
k
/C
k
)].
Write
μ
−
∈
X
k
(k)
28
SHINICHI
MOCHIZUKI
for
the
unique
torsion
point
of
order
2
whose
closure
in
any
stable
model
of
X
k
over
O
k
intersects
the
same
irreducible
component
of
the
special
fiber
of
the
stable
model
as
the
zero
cusp
[cf.
the
discussion
of
[IUTchI],
Example
4.4,
(i)].
(ii)
The
unique
order
two
automorphism
ι
X
of
X
k
over
k
[cf.
[EtTh],
Remark
2.6.1]
lies
over
an
order
two
automorphism
ι
X
[cf.
[EtTh],
Remark
2.6.1]
and
corresponds
at
the
level
of
tempered
fundamental
groups
[cf.,
e.g.,
[SemiAnbd],
tp
Theorem
6.4]
to
the
unique
order
two
Δ
tp
X
-outer
automorphism
of
Π
X
over
G
k
,
k
k
which,
by
abuse
of
notation,
we
shall
also
denote
by
ι
X
.
Write
Ÿ
k
→
Y
k
→
X
k
for
the
tempered
coverings
of
X
k
that
correspond,
respectively,
to
the
open
sub-
def
groups
Π
tp
Ÿ
Π
Y
(M
Θ
def
tp
tp
=
Π
Ÿ
(Π
tp
X
)
⊆
Π
X
[cf.
Proposition
1.4],
Π
Y
k
k
tp
(Π
tp
X
))
⊆
Π
X
k
k
def
=
Π
Y
(Π
tp
X
)
=
k
k
[cf.
Definition
1.1,
(i);
Proposition
1.2,
(i)].
Since
k
con-
k
tains
a
primitive
4l-th
root
of
unity,
it
follows
from
the
definition
of
an
“étale
theta
function
of
standard
type”
[cf.
[EtTh],
Definition
1.9,
(ii);
[EtTh],
Definition
2.7]
that
there
exist
rational
points
(μ
−
)
Ÿ
∈
Ÿ
k
(k),
(μ
−
)
X
∈
X
k
(k)
such
that
(μ
−
)
Ÿ
→
(μ
−
)
X
→
μ
−
.
Since
ι
X
fixes
μ
−
,
it
follows
immediately
that
ι
X
fixes
the
Gal(X
/X
)-orbit
of
(μ
−
)
X
,
hence
[since
Aut(X
)
∼
=
Z/2lZ,
where
we
k
k
k
recall
that
l
=
2
—
cf.
[EtTh],
Remark
2.6.1]
that
ι
X
fixes
(μ
−
)
X
.
One
verifies
immediately
that
this
implies
that
there
exists
an
order
two
automorphism
ι
Ÿ
of
Ÿ
k
lifting
ι
X
which
is
uniquely
determined
up
to
l
·
Z-conjugacy
and
composition
with
an
element
∈
Gal(
Ÿ
k
/Y
k
)
by
the
condition
that
it
fix
the
Gal(
Ÿ
k
/Y
k
)-orbit
of
some
element
[which,
by
abuse
of
notation,
we
shall
continue
to
denote
by
“(μ
−
)
Ÿ
”]
of
the
Gal(
Ÿ
/X
)-orbit
of
(μ
−
)
.
Here,
we
think
of
l
·
Z,
Gal(
Ÿ
/Y
)
(
∼
=
Z/2Z)
k
k
Ÿ
k
k
as
the
subquotients
appearing
in
the
natural
exact
sequence
1
→
Gal(
Ÿ
k
/Y
k
)
→
Gal(
Ÿ
k
/X
k
)
→
l
·
Z
→
1
determined
by
the
coverings
Ÿ
k
→
Y
k
→
X
k
.
Again,
by
abuse
of
notation,
we
Π
tp
shall
also
denote
by
ι
Ÿ
the
corresponding
Δ
tp
(=
Δ
tp
)-outer
automor-
X
Ÿ
Ÿ
phism
of
Π
tp
.
Ÿ
k
k
k
k
We
shall
refer
to
the
various
automorphisms
ι
X
,
ι
Ÿ
as
inversion
automorphisms
[cf.
[EtTh],
Proposition
1.5,
(iii)].
Write
D
μ
−
⊆
Π
Ÿ
k
for
the
decomposition
group
of
(μ
−
)
Ÿ
[which
is
well-defined
up
to
Δ
tp
-conjugacy]
Ÿ
k
def
tp
Θ
—
so
D
μ
−
is
determined
by
ι
Ÿ
up
to
Δ
tp
Y
k
(
=
Δ
Y
(M
(Π
X
)))-conjugacy
[cf.
the
k
notation
of
Remark
1.1.1,
(i)].
We
shall
refer
to
either
of
the
pairs
(ι
Ÿ
∈
Aut(
Ÿ
k
),
(μ
−
)
Ÿ
);
(ι
Ÿ
∈
Aut(Π
tp
)/Inn(Δ
tp
),
D
μ
−
)
Ÿ
Ÿ
k
k
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
29
as
a
pointed
inversion
automorphism.
Again,
we
recall
from
[EtTh],
Definition
1.9,
(ii);
[EtTh],
Definition
2.7,
that
an
“étale
theta
function
of
standard
type”
is
defined
precisely
by
the
con-
dition
that
its
restriction
to
D
μ
−
be
a
2l-th
root
of
unity.
Proposition
1.5.
(Projective
Systems
of
Mono-theta
Environments)
In
the
notation
of
the
above
discussion,
let
M
Θ
∗
=
Θ
{.
.
.
→
M
Θ
M
→
M
M
→
.
.
.
}
be
a
projective
system
of
mono-theta
environments
—
where
M
Θ
M
is
a
mod
M
mono-theta
environment
[which
is
isomorphic
to
the
mod
M
model
mono-theta
environment
determined
by
X
k
],
and
the
index
M
of
the
projective
system
varies
multiplicatively
among
the
elements
of
N
≥1
[cf.
[EtTh],
Corollary
2.19,
(ii),
(iii)].
Then:
(i)
Such
a
projective
system
is
uniquely
determined,
up
to
isomorphism,
by
X
k
[cf.
Remark
1.5.1
below;
the
discrete
rigidity
property
of
[EtTh],
Corollary
2.19,
(ii)].
(ii)
The
transition
morphisms
of
the
resulting
projective
system
of
topological
Θ
groups
{.
.
.
→
Π
X
(M
Θ
M
)
→
Π
X
(M
M
)
→
.
.
.
}
[cf.
the
notation
of
Definition
1.1,
(i)]
are
all
isomorphisms.
Moreover,
any
isomorphism
of
topological
groups
∼
Θ
Π
X
(M
Θ
M
)
→
Π
X
(M
M
),
where
M
divides
M
,
lifts
to
a
morphism
of
mono-theta
Θ
environments
M
Θ
M
→
M
M
[cf.
[EtTh],
Corollary
2.18,
(iv)].
Thus,
to
simplify
the
notation,
we
shall
identify
these
topological
groups
via
these
transition
morphisms
and
denote
the
resulting
topological
group
by
the
notation
Π
X
(M
Θ
∗
).
In
particular,
Θ
Θ
we
have
an
open
subgroup
Π
Ÿ
(M
Θ
∗
)
⊆
Π
X
(M
∗
),
a
subquotient
(l
·Δ
Θ
)(M
∗
)
of
Θ
Θ
Π
X
(M
Θ
∗
),
and
a
quotient
Π
X
(M
∗
)
G(M
∗
)
[cf.
Definition
1.1,
(i);
Proposition
1.4].
→
(iii)
The
projective
system
of
exterior
cyclotomes
{.
.
.
→
Π
μ
(M
Θ
M
)
Θ
Π
μ
(M
M
)
→
.
.
.
}
[cf.
the
notation
of
Definition
1.1,
(i)]
determines
a
projective
limit
exterior
cyclotome
Π
μ
(M
Θ
∗
)
which
is
equipped
with
a
uniquely
determined
cyclotomic
rigidity
isomorphism
∼
Θ
(l
·
Δ
Θ
)(M
Θ
∗
)
→
Π
μ
(M
∗
)
[i.e.,
obtained
by
applying
the
cyclotomic
rigidity
isomorphisms
of
Definition
1.1,
(ii),
to
the
various
members
of
the
projective
system
M
Θ
∗
].
In
particular,
[cf.
Propo-
sition
1.4]
we
obtain
a
functorial
algorithm
M
Θ
∗
→
θ
env
(M
Θ
∗
)
⊆
Θ
H
1
(Π
Ÿ
(M
Θ
∗
),
Π
μ
(M
∗
))
—
where
one
may
think
of
the
“env”
as
an
abbreviation
of
the
term
“[mono-theta]
environment”
—
for
constructing
from
M
Θ
∗
an
exterior
cyclotome
version
30
SHINICHI
MOCHIZUKI
θ
env
(M
Θ
∗
)
of
θ(Π)
[i.e.,
by
transporting
θ(Π)
via
the
above
cyclotomic
rigidity
iso-
morphism]
—
cf.
[EtTh],
Corollary
2.19,
(iii).
In
this
context,
we
shall
write
Θ
∞
θ
env
(M
∗
)
⊆
1
Θ
Θ
lim
−→
H
(Π
Ÿ
(M
∗
)|
J
,
Π
μ
(M
∗
))
J
—
where
∞
θ
env
(M
Θ
∗
)
denotes
the
subset
of
elements
of
the
direct
limit
of
cohomol-
ogy
modules
in
the
display
for
which
some
[positive
integer]
multiple
[i.e.,
some
[positive
integer]
power,
if
one
writes
these
modules
“multiplicatively”]
coincides,
up
to
torsion,
with
an
element
of
θ
env
(M
Θ
∗
);
J
ranges
over
the
finite
index
open
Θ
subgroups
of
Π
X
(M
∗
).
(iv)
Suppose
that
M
Θ
∗
arises
from
a
tempered
Frobenioid
C
[cf.
Propositions
1.2,
(ii);
1.3].
Then
this
construction
of
θ
env
(M
Θ
∗
)
[cf.
(iii)]
is
compatible
with
the
Kummer-theoretic
construction
of
the
étale
theta
function
—
i.e.,
by
con-
sidering
Galois
actions
on
roots
of
the
Frobenioid-theoretic
theta
function
[cf.
the
theory
of
[EtTh],
§5].
In
particular,
it
is
compatible
with
the
Kummer
theory
of
the
base-field-theoretic
hull
C
bs-fld
[cf.
[FrdII],
Theorem
2.4;
[AbsTopIII],
Proposition
3.2,
(ii);
[AbsTopIII],
Remark
3.1.1].
Proof.
The
assertions
of
Proposition
1.5
follow
immediately
from
the
results
and
definitions
of
[EtTh]
[as
well
as
[FrdII],
[AbsTopIII]]
that
are
quoted
in
the
state-
ments
of
these
assertions.
Remark
1.5.1.
We
recall
in
passing
that
one
important
consequence
of
the
discrete
rigidity
property
established
in
[EtTh],
Corollary
2.19,
(ii)
—
which,
in
effect,
allows
one
to
restrict
one’s
attention
to
l
·
Z-translates
[i.e.,
as
opposed
to
l
·
Z-translates]
of
the
usual
theta
function
—
is
the
resulting
compatibility
of
projective
systems
of
mono-theta
environments
[as
in
Proposition
1.5]
with
the
discrete
structure
inherent
in
the
various
isomorphs
of
the
monoid
N
that
appear
in
the
structure
of
the
tempered
Frobenioids
that
arise
in
the
theory
[cf.
[EtTh],
Remark
2.19.4;
[EtTh],
Remark
5.10.4,
(i),
(ii)].
Remark
1.5.2.
Note
that,
in
the
notation
of
Proposition
1.5,
(iii),
by
consider-
ing
“tautological
Kummer
classes”
of
elements
of
Π
μ
(M
Θ
∗
),
one
obtains
a
natural
Θ
Π
X
(M
∗
)-equivariant
injection
1
Θ
Θ
Π
μ
(M
Θ
∗
)
⊗
Q/Z
→
lim
−→
H
(Π
Ÿ
(M
∗
)|
J
,
Π
μ
(M
∗
))
J
whose
image
is
equal
to
the
torsion
subgroup
of
the
codomain
of
the
injection.
Indeed,
it
follows
immediately
from
the
fact
that
Π
μ
(M
Θ
∗
)
is
torsion-free
that
the
torsion
subgroup
of
the
codomain
of
the
displayed
injection
may
be
identified
with
the
torsion
subgroup
of
1
Θ
lim
−→
H
(J
G
,
Π
μ
(M
∗
))
J
—
where
J
ranges
over
the
finite
index
open
subgroups
of
Π
X
(M
Θ
∗
);
we
write
J
G
for
the
image
of
J
in
G(M
Θ
∗
).
The
desired
conclusion
thus
follows
immediately
from
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
31
the
well-known
Kummer
theory
of
MLF’s,
i.e.,
the
fact
that
the
Kummer
map
J
→
H
1
(J
G
,
Π
μ
(M
Θ
(Π
μ
(M
Θ
∗
)
⊗
Q/Z)
∗
))
[where
the
superscript
“J”
denotes
the
submodule
of
J-invariants]
is
injective
with
image
equal
to
the
torsion
subgroup
of
the
codomain.
Before
proceeding,
we
review
a
certain
portion
of
the
theory
of
[AbsTopII]
that
is
relevant
to
the
content
of
the
present
§1.
Proposition
1.6.
(Cores
and
Cuspidalizations)
Let
Π
be
as
in
Proposition
1.2,
(i).
Write
Δ
⊆
Π
for
the
[group-theoretic!
—
cf.,
e.g.,
[AbsAnab],
Lemma
1.3.8]
subgroup
corresponding
to
Δ
tp
X
.
Then:
k
(i)
(Cores)
There
exists
a
functorial
group-theoretic
algorithm
[cf.
[Ab-
sTopII],
Corollary
3.3,
(i);
[AbsTopII],
Remark
3.3.3]
Π
→
(Π
⊆)
Π
C
(Π)
Π/Δ
for
constructing
from
Π
a
topological
group
Π
C
(Π)
equipped
with
an
augmentation
[i.e.,
a
surjection]
Π
C
(Π)
Π/Δ
—
whose
kernel
we
denote
by
Δ
C
(Π)
—
that
contains
Π
as
an
open
subgroup
in
a
fashion
that
is
compatible
with
the
respec-
tive
surjections
to
Π/Δ
and
which
satisfies
the
property
that
when
Π
=
Π
tp
X
,
the
k
tp
inclusion
Π
⊆
Π
C
(Π)
may
be
naturally
identified
with
the
inclusion
Π
tp
X
⊆
Π
C
k
.
k
(ii)
(Elliptic
Cuspidalizations)
Let
N
be
a
positive
integer.
Then
there
exists
a
functorial
group-theoretic
algorithm
[cf.
[AbsTopII],
Corollary
3.3,
(iii);
[AbsTopII],
Remark
3.3.3]
Π
→
Π
U
N
(Π)
Π
for
constructing
from
Π
a
topological
group
Π
U
N
(Π)
equipped
with
a
surjection
Π
U
N
(Π)
Π
[so
the
augmentation
Π
Π/Δ
determines,
by
composition,
an
aug-
mentation
Π
U
N
(Π)
Π/Δ]
such
that
when
Π
=
Π
tp
X
,
the
surjection
Π
U
N
(Π)
Π
k
may
be
naturally
identified
with
a
certain
surjection
—
i.e.,
“elliptic
cuspidaliza-
tion”
—
that
arises
from
a
certain
open
immersion
determined
by
the
N
-torsion
points
of
a
once-punctured
elliptic
curve
that
forms
a
double
covering
of
C
k
[cf.
[AbsTopII],
Corollary
3.3,
(iii)].
Proof.
The
assertions
of
Proposition
1.6
follow
immediately
from
the
results
of
[AbsTopII]
that
are
quoted
in
the
statements
of
these
assertions
[cf.
also
Remark
1.6.1
below].
Remark
1.6.1.
We
recall
in
passing
that
the
construction
of
Proposition
1.6,
(i),
amounts,
in
effect,
to
the
computation
of
various
centralizers
of
the
image
of
various
open
subgroups
of
Π/Δ
in
the
outer
automorphism
groups
of
various
open
subgroups
of
Δ.
In
a
similar
vein,
the
construction
of
Proposition
1.6,
(ii),
amounts
to
the
computation
of
various
outer
isomorphisms
between
various
subquotients
of
32
SHINICHI
MOCHIZUKI
Δ
that
are
compatible
with
the
outer
actions
of
various
open
subgroups
of
Π/Δ.
More
generally,
although
in
Proposition
1.6,
we
restricted
our
attention
to
the
con-
struction
of
cores
and
elliptic
cuspidalizations,
an
analogous
result
may
be
obtained
for
more
general
functorial
group-theoretic
algorithms
involving
“chains
of
elemen-
tary
operations”,
as
discussed
in
[AbsTopI],
§4
—
e.g.,
for
Belyi
cuspidalizations,
as
discussed
in
[AbsTopII],
Corollary
3.7.
Next,
we
proceed
to
discuss
the
“multiradial”
interpretation
of
the
theory
of
[EtTh]
that
is
of
interest
in
the
context
of
the
present
series
of
papers.
We
begin
by
examining
various
examples
of
the
sort
of
situation
that
gives
rise
to
such
an
interpretation.
Example
1.7.
Radial
and
Coric
Data
I:
Generalities.
(i)
In
the
following
discussion,
we
would
like
to
consider
a
certain
“type
of
mathematical
data”,
which
we
shall
refer
to
as
radial
data.
This
notion
of
a
“type
of
mathematical
data”
may
be
formalized
—
cf.
[IUTchIV],
§3,
for
more
details.
From
the
point
of
view
of
the
present
discussion,
one
may
think
of
a
“type
of
mathematical
data”
as
the
input
or
output
data
of
a
“functorial
algorithm”
[cf.
the
discussion
of
[IUTchI],
Remark
3.2.1].
At
a
more
concrete
level,
we
shall
assume
that
this
“type
of
mathematical
data”
gives
rise
to
a
category
R
—
i.e.,
each
of
whose
objects
is
a
specific
collection
of
radial
data,
and
each
of
whose
morphisms
is
an
isomorphism.
In
the
following
discussion,
we
shall
also
consider
another
“type
of
mathematical
data”,
which
we
shall
refer
to
as
coric
data.
Write
C
for
the
category
obtained
by
considering
specific
collections
of
coric
data
and
iso-
morphisms
of
collections
of
coric
data.
In
addition,
we
shall
assume
that
we
are
given
a
functorial
algorithm
—
which
we
shall
refer
to
as
radial
—
whose
input
data
consists
of
a
collection
of
radial
data,
and
whose
output
data
consists
of
a
collection
of
coric
data.
Thus,
this
functorial
algorithm
gives
rise
to
a
functor
Φ
:
R
→
C.
In
the
following
discussion,
we
shall
assume
that
this
functor
is
essentially
surjective.
We
shall
refer
to
the
category
R
and
the
functor
Φ
as
radial
and
to
the
category
C
as
coric.
Finally,
if
I
is
some
nonempty
index
set,
then
we
shall
often
consider
collections
{Φ
i
:
R
i
→
C}
i∈I
of
copies
of
Φ
and
R,
such
that
the
various
copies
of
Φ
have
the
same
codomain
C
—
cf.
Fig.
1.1
below.
Thus,
one
may
think
of
each
R
i
as
the
category
of
radial
data
equipped
with
a
label
i
∈
I,
and
isomorphisms
of
such
data.
(ii)
We
shall
refer
to
a
triple
(R,
C,
Φ
:
R
→
C)
[or
to
the
triple
consisting
of
the
corresponding
“types
of
mathematical
objects”
and
“functorial
algorithm”]
of
the
sort
discussed
in
(i)
as
a
radial
environment.
If
Φ
is
full,
then
we
shall
refer
to
the
radial
environment
under
consideration
as
multiradial.
We
shall
refer
to
a
radial
environment
which
is
not
multiradial
as
uniradial.
Suppose
that
the
radial
environment
(R,
C,
Φ
:
R
→
C)
under
consideration
is
uniradial.
Then
an
object
of
R
may,
in
general,
lose
a
certain
portion
of
its
rigidity
—
i.e.,
may
be
subject
to
a
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
33
certain
additional
indeterminacy
—
when
it
is
mapped
to
C.
Put
another
way,
in
general,
an
object
of
C
is
imparted
with
a
certain
additional
rigidity
—
i.e.,
loses
a
certain
portion
of
its
indeterminacy
—
when
one
fixes
a
lifting
of
the
object
to
R.
Thus,
in
summary,
the
condition
that
(R,
C,
Φ
:
R
→
C)
be
multiradial
may
be
thought
of
as
a
condition
to
the
effect
that
the
application
of
the
radial
algorithm
does
not
result
in
any
loss
of
rigidity.
Finally,
we
observe
that,
if
(R,
C,
Φ
:
R
→
C)
is
an
arbitrary
radial
environment
such
that
any
two
collections
of
radial
data
are
isomorphic,
then
one
may
define
the
associated
[tautological]
multiradialization
(R
mtz
,
C,
Φ
mtz
:
R
mtz
→
C)
of
this
radial
environment
as
follows:
A
collection
of
radial
data
(R,
C,
α)
of
this
multiradialization
consists
of
an
object
R
of
R,
an
object
C
of
C,
and
the
full
∼
poly-isomorphism
[cf.
[IUTchI],
§0]
α
:
Φ(R)
→
C.
An
isomorphism
of
collections
∼
of
radial
data
(R,
C,
α)
→
(R
∗
,
C
∗
,
α
∗
)
of
the
multiradialization
consists
of
a
pair
of
∼
∼
isomorphisms
R
→
R
∗
,
C
→
C
∗
[which
are
necessarily
compatible
with
α,
α
∗
].
The
coric
data
of
the
multiradialization
is
taken
to
be
the
coric
data
of
the
original
radial
environment
(R,
C,
Φ
:
R
→
C).
The
radial
algorithm
of
the
multiradialization
is
taken
to
be
the
assignment
(R,
C,
α)
→
C
—
whose
associated
radial
functor
is
clearly
full
[cf.
our
assumption
that
any
two
collections
of
radial
data
are
isomorphic!]
and
essentially
surjective,
hence
determines
a
[tautologically!]
multiradial
environment
(R
mtz
,
C,
Φ
mtz
:
R
mtz
→
C),
together
with
a
natural
functor
R
→
R
mtz
[i.e.,
given
by
the
assignment
R
→
∼
(R,
Φ(R),
Φ(R)
→
Φ(R))].
Indeed,
the
tautological
multiradialization
of
the
given
radial
environment
may
be
thought
of
as
the
result
of
“forgetting,
in
a
minimal
possible
fash-
ion,
the
uniradiality”
of
the
original
radial
environment
(R,
C,
Φ
:
R
→
C).
R
i
...
R
i
⏐
⏐
...
−→
C
←−
R
i
...
⏐
⏐
...
R
i
Fig.
1.1:
Radial
functors
valued
in
a
single
coric
category
34
SHINICHI
MOCHIZUKI
(iii)
In
passing,
we
pause
to
observe
that
one
way
to
think
of
the
significance
of
the
multiradiality
of
a
radial
environment
(R,
C,
Φ
:
R
→
C)
is
as
follows:
Write
R
×
C
R
for
the
category
whose
objects
are
triples
(R
1
,
R
2
,
α)
consisting
of
a
pair
of
objects
∼
R
1
,
R
2
of
R
and
an
isomorphism
α
:
Φ(R
1
)
→
Φ(R
2
)
between
the
images
of
R
1
,
R
2
via
Φ,
and
whose
morphisms
are
the
morphisms
[in
the
evident
sense]
between
such
triples
[cf.
the
discussion
of
the
“categorical
fiber
product”
given
in
[FrdI],
§0].
∼
Write
sw
:
R
×
C
R
→
R
×
C
R
for
the
functor
(R
1
,
R
2
,
α)
→
(R
2
,
R
1
,
α
−1
)
obtained
by
switching
the
two
factors
of
R.
Then
one
formal
consequence
of
the
multiradiality
of
a
radial
environment
(R,
C,
Φ
:
R
→
C)
is
the
property
that
the
switching
functor
sw
:
∼
R×
C
R
→
R×
C
R
preserves
the
isomorphism
class
of
objects
of
R×
C
R.
Indeed,
one
verifies
immediately
that
this
multiradiality
is,
in
fact,
equivalent
to
the
condition
that
every
object
(R
1
,
R
2
,
α)
of
R
×
C
R
be
isomorphic
to
the
object
∼
(R
1
,
R
1
,
id
:
Φ(R
1
)
→
Φ(R
1
))
[which
is
manifestly
left
unchanged
by
the
switching
functor].
(iv)
Next,
suppose
that
we
are
given
another
radial
environment
(R
†
,
C
†
,
Φ
†
:
R
†
→
C
†
).
We
shall
refer
to
the
“type
of
mathematical
object”/“functorial
algo-
rithm”
that
gives
rise
to
R
†
(respectively,
C
†
;
Φ
†
)
as
daggered
radial
data
(respec-
tively,
daggered
coric
data;
the
daggered
radial
functorial
algorithm).
Also,
let
us
suppose
that
we
are
given
a
1-commutative
diagram
Ψ
R
R
−→
R
†
⏐
⏐
⏐
⏐
†
Φ
Φ
C
Ψ
C
−→
C
†
—
where
Ψ
R
and
Ψ
C
arise
from
“functorial
algorithms”.
If
(R,
C,
Φ
:
R
→
C)
is
multiradial
(respectively,
uniradial),
then
we
shall
refer
to
Ψ
R
as
multiradially
defined
(respectively,
uniradially
defined),
or
[when
there
is
no
fear
of
confusion
between
Φ
and
Ψ
R
]
as
multiradial
(respectively,
uniradial).
If
Ψ
R
admits
a
1-
factorization
Ξ
R
◦
Φ
for
some
Ξ
R
:
C
→
R
†
that
arises
from
a
functorial
algorithm,
then
we
shall
say
that
Ψ
R
is
corically
defined,
or
[when
there
is
no
fear
of
confusion]
coric.
Thus,
by
considering
the
case
where
R
=
C,
Φ
=
id
R
,
one
may
think
of
the
notion
of
a
corically
defined
Ψ
R
as
a
sort
of
special
case
of
the
notion
of
a
multiradial
Ψ
R
.
(v)
Suppose
that
we
are
in
the
situation
of
(iv),
and
that
Ψ
R
is
multiradially
defined.
Then
one
way
to
think
of
the
significance
of
the
multiradiality
of
Ψ
R
is
as
follows:
The
multiradiality
of
Ψ
R
renders
it
possible
to
consider
the
simultaneous
execution
of
the
functorial
algorithm
corresponding
to
Ψ
R
relative
to
various
collections
of
radial
input
data
indexed
by
the
set
I
[cf.
Fig.
1.1]
in
a
fashion
that
is
compatible
with
the
identification
of
the
coric
portions
[i.e.,
corresponding
to
Φ]
of
these
collections
of
radial
input
data
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
35
—
cf.
Remark
1.9.1
below
for
more
on
this
point
of
view.
That
is
to
say,
at
a
more
technical
level,
if
one
implements
this
identification
of
the
various
coric
portions
by
means
of
various
gluing
isomorphisms
in
C,
then
the
multiradiality
of
Ψ
R
implies
that
one
may
lift
these
gluing
isomorphisms
in
C
to
gluing
isomorphisms
in
R;
one
may
then
apply
Ψ
R
to
these
gluing
isomorphisms
in
R
to
obtain
gluing
isomor-
phisms
of
the
output
data
of
Ψ
R
.
Put
another
way,
if
one
assumes
instead
that
Ψ
R
is
uniradial,
then
the
output
data
of
Ψ
R
depends,
a
priori,
on
the
“additional
rigidity”
[cf.
(ii)]
of
objects
of
R
relative
to
these
images
in
C;
thus,
if
one
attempts
to
identify
these
images
in
C
via
arbitrary
gluing
isomorphisms
in
C,
then
one
does
not
have
any
way
to
compute
the
effect
of
such
gluing
isomorphisms
on
the
output
data
of
Ψ
R
.
Remark
1.7.1.
One
way
to
understand
the
significance
of
the
fullness
condi-
tion
in
the
definition
of
a
multiradial
environment
is
as
a
condition
that
allows
one
to
execute
a
sort
of
parallel
transport
operation
between
“fibers”
of
the
ra-
dial
functor
Φ
:
R
→
C
[cf.
the
notation
of
Example
1.7,
(iv)]
—
i.e.,
by
lifting
isomorphisms
in
C
to
isomorphisms
in
R
[cf.
the
discussion
of
Example
1.7,
(v)].
Here,
it
is
perhaps
of
interest
to
make
the
tautological
observation
that,
up
to
an
indeterminacy
arising
from
the
extent
that
Φ
fails
to
be
faithful,
such
liftings
are
unique.
That
is
to
say,
whereas
a
uniradial
environment
may
be
thought
of
as
a
sort
of
abstraction
of
the
geometric
notion
of
a
“fibration
that
is
not
equipped
with
a
connection”,
a
multiradial
environment
may
be
thought
of
as
a
sort
of
abstraction
of
the
geometric
notion
of
a
“fibration
equipped
with
a
connection”
—
i.e.,
that
allows
one
to
execute
parallel
transport
operations
between
the
“fibers”.
Relative
to
this
point
of
view,
one
may
think
of
the
coric
data
as
the
portion
of
the
radial
data
of
a
multiradial
environment
that
is
horizontal
with
respect
to
the
“connection
structure”.
We
refer
to
Remarks
1.9.1,
1.9.2
below
for
more
on
the
significance
of
multiradiality.
Example
1.8.
Radial
and
Coric
Data
II:
Concrete
Examples.
In
this
following,
we
consider
various
concrete
examples
of
multiradial
environments,
many
of
which
may,
in
fact,
be
understood
as
special
cases
of
the
notion
of
the
tautological
multiradialization
associated
to
a
suitable
choice
of
radial
environment,
i.e.,
as
discussed
in
Example
1.7,
(ii).
(i)
From
the
point
of
view
of
the
theory
to
be
developed
in
the
remainder
of
the
present
§1,
perhaps
the
most
basic
example
of
a
radial
environment
is
the
following.
We
define
a
collection
of
radial
data
(Π,
G,
α)
to
consist
of
a
topological
group
Π
isomorphic
to
Π
tp
X
,
a
topological
group
G
iso-
k
morphic
to
G
k
,
and
the
full
poly-isomorphism
[cf.
[IUTchI],
§0]
of
topological
∼
groups
α
:
Π/Δ
→
G,
where
we
write
Δ
⊆
Π
for
the
[group-theoretic!
—
cf.,
e.g.,
[AbsAnab],
Lemma
1.3.8]
subgroup
corresponding
to
Δ
tp
X
.
An
isomorphism
k
36
SHINICHI
MOCHIZUKI
∼
of
collections
of
radial
data
(Π,
G,
α)
→
(Π
∗
,
G
∗
,
α
∗
)
is
defined
to
be
a
pair
of
∼
∼
isomorphisms
of
topological
groups
Π
→
Π
∗
,
G
→
G
∗
[which
are
necessarily
com-
patible
with
α,
α
∗
!].
A
collection
of
coric
data
is
defined
to
be
a
topological
group
isomorphic
to
G
k
;
an
isomorphism
of
collections
of
coric
data
is
defined
to
be
an
isomorphism
of
topological
groups.
The
radial
algorithm
is
the
algorithm
given
by
the
assignment
(Π,
G,
α)
→
G
—
whose
associated
radial
functor
is
full
and
essentially
surjective,
hence
determines
a
multiradial
environment.
Note
that
this
example
may
be
thought
of
as
a
sort
of
formalization
in
the
present
context
of
the
situation
depicted
in
[IUTchI],
Fig.
3.2,
at
v
∈
V
bad
—
cf.
Fig.
1.2
below.
Here,
we
recall
that
the
topological
group
“G”
[which
is
isomorphic
to
G
k
]
that
appears
in
the
center
of
Fig.
1.2
is
regarded
as
being
known
only
up
to
isomorphism,
and
that
the
various
isomorphs
of
Π
X
k
that
appear
in
the
“spokes”
of
Fig.
1.2
may
be
regarded
as
various
“arithmetic
holomorphic
structures”
on
“G”
[cf.
[IUTchI],
Remark
3.8.1,
(iii)].
i
Π
⏐
⏐
...
Π
−→
G
←−
...
⏐
⏐
...
...
i
i
i
Π
Π
Fig.
1.2:
Different
arithmetic
holomorphic
structures
on
a
single
coric
G
(ii)
Recall
the
functorial
group-theoretic
algorithm
Π
→
(Π
M
TM
(Π))
(∗
TM
)
of
[AbsTopIII],
§3
[cf.,
especially,
the
functors
κ
An
,
φ
An
of
[AbsTopIII],
Definition
3.1,
(vi);
[AbsTopIII],
Corollary
3.6,
(ii);
[IUTchI],
Remark
3.1.2]
that
assigns
to
a
topological
group
Π
isomorphic
to
Π
tp
X
an
MLF-Galois
TM-pair,
which
we
shall
k
denote
Π
M
TM
(Π),
and
which
is
isomorphic
to
the
“model”
MLF-Galois
TM-
pair
determined
by
the
natural
action
of
Π
tp
X
on
the
ind-topological
monoid
O
.
In
k
k
fact,
[the
union
with
{0}
of]
the
underlying
ind-topological
monoid
M
TM
(Π)
is
also
equipped
with
a
natural
ring
structure
[cf.
[AbsTopIII],
Proposition
3.2,
(iii)].
On
the
other
hand,
if
one
is
willing
to
sacrifice
this
ring
structure,
then
there
exists
a
functorial
group-theoretic
algorithm
G
→
(G
O
(G))
(∗
)
[cf.
[AbsTopIII],
Proposition
5.8,
(i)]
that
assigns
to
a
topological
group
G
isomor-
phic
to
G
k
an
MLF-Galois
TM-pair,
which
we
shall
denote
G
O
(G),
and
which
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
37
is
isomorphic
to
the
MLF-Galois
TM-pair
determined
by
the
natural
action
of
G
k
on
the
ind-topological
monoid
O
.
Moreover,
by
[AbsTopIII],
Proposition
3.2,
(iv)
k
[cf.
also
Remark
1.11.1,
(i),
(a),
below],
there
is
a
[uniquely
determined]
functorial
tautological
isomorphism
of
MLF-Galois
TM-pairs
∼
(Π/Δ
O
(Π/Δ))|
Π
→
(Π
M
TM
(Π))
(∗
TM
)
—
where
Δ
⊆
Π
is
as
in
(i),
and
the
notation
“|
Π
”
denotes
the
restriction
of
the
action
of
Π/Δ
to
an
action
of
Π.
Then
another
important
example
of
a
radial
environment
is
the
following.
We
define
a
collection
of
radial
data
(Π
M
TM
(Π),
G
O
(G),
α
)
to
consist
of
the
output
data
of
the
algorithm
(∗
TM
)
associated
to
a
topological
group
Π
isomorphic
to
Π
tp
X
,
the
output
data
of
the
algorithm
(∗
)
associated
to
a
k
topological
group
G
isomorphic
to
G
k
,
and
the
poly-isomorphism
[cf.
[IUTchI],
§0]
of
MLF-Galois
TM-pairs
∼
α
:
(Π
M
TM
(Π))
(G
O
(G))|
Π
→
determined
[in
light
of
[AbsTopIII],
Proposition
3.2,
(iv)]
by
the
composite
of
the
natural
surjection
Π
Π/Δ
with
the
full
poly-isomorphism
of
topological
groups
∼
Π/Δ
→
G
[where
Δ
⊆
Π
is
as
in
(i)].
An
isomorphism
of
collections
of
radial
data
∼
∗
(Π
M
TM
(Π),
G
O
(G),
α
)
→
(Π
∗
M
TM
(Π
∗
),
G
∗
O
(G
∗
),
α
)
is
de-
∼
fined
to
be
a
pair
of
isomorphisms
of
MLF-Galois
TM-pairs
(Π
M
TM
(Π))
→
(Π
∗
∼
M
TM
(Π
∗
)),
(G
O
(G))
→
(G
∗
O
(G
∗
))
[which
are
necessarily
compatible
∗
!].
A
collection
of
coric
data
is
defined
to
be
the
output
data
of
the
with
α
,
α
algorithm
(∗
)
for
some
topological
group
isomorphic
to
G
k
;
an
isomorphism
of
collections
of
coric
data
is
defined
to
be
the
isomorphism
between
collections
of
output
data
of
(∗
)
associated
to
an
isomorphism
of
topological
groups.
The
ra-
dial
algorithm
is
the
algorithm
given
by
the
assignment
(Π
M
TM
(Π),
G
O
(G),
α
)
→
(G
O
(G))
—
whose
associated
radial
functor
is
full
and
essentially
surjective,
hence
determines
a
multiradial
environment.
(iii)
Let
Γ
⊆
Z
×
be
a
closed
subgroup
[cf.
Remark
1.11.1,
(i),
(ii),
below,
for
more
on
the
significance
of
Γ].
Then
by
considering
the
subgroups
of
invertible
elements
of
the
various
ind-
topological
monoids
that
appeared
in
(ii),
one
obtains
functorial
group-theoretic
algorithms
Π
→
×
(Π));
(Π
M
TM
G
→
(G
O
×
(G))
(∗
×
)
defined,
respectively,
on
topological
groups
Π
isomorphic
to
Π
tp
X
and
G
isomorphic
k
to
G
k
.
Here,
we
note
that
we
may
think
of
Γ
as
acting
on
the
output
data
of
the
38
SHINICHI
MOCHIZUKI
second
algorithm
of
(∗
×
)
by
means
of
the
trivial
action
on
G
and
the
natural
action
of
Z
×
on
O
×
(G).
Then
one
obtains
another
example
of
a
radial
environment
as
follows.
We
define
a
collection
of
radial
data
×
(Π),
G
O
×
(G),
α
×
)
(Π
M
TM
to
consist
of
the
output
data
of
the
first
algorithm
of
(∗
×
)
associated
to
a
topolog-
ical
group
Π
isomorphic
to
Π
tp
X
,
the
output
data
of
the
second
algorithm
of
(∗
×
)
k
associated
to
a
topological
group
G
isomorphic
to
G
k
,
and
the
poly-isomorphism
[cf.
[IUTchI],
§0]
of
ind-topological
modules
equipped
with
topological
group
actions
×
α
×
:
(Π
M
TM
(Π))
∼
(G
O
×
(G))|
Π
→
determined
by
the
Γ-orbit
of
the
poly-isomorphism
“α
|
×
”
induced
by
the
poly-
×
isomorphism
α
of
(ii).
An
isomorphism
of
collections
of
radial
data
(Π
M
TM
(Π),
∼
×
×
∗
∗
∗
×
∗
∗
G
O
(G),
α
×
)
→
(Π
M
TM
(Π
),
G
O
(G
),
α
×
)
is
defined
to
consist
of
the
isomorphism
of
ind-topological
modules
equipped
with
topological
group
actions
∼
×
×
(Π))
→
(Π
∗
M
TM
(Π
∗
))
induced
by
an
isomorphism
of
topological
(Π
M
TM
∼
groups
Π
→
Π
∗
,
together
with
a
Γ-multiple
of
the
isomorphism
of
ind-topological
∼
modules
equipped
with
topological
group
actions
(G
O
×
(G))
→
(G
∗
O
×
(G
∗
))
∼
induced
by
an
isomorphism
of
topological
groups
G
→
G
∗
[so
one
verifies
immedi-
∗
in
the
evident
sense].
A
ately
that
these
isomorphisms
are
compatible
with
α
×
,
α
×
collection
of
coric
data
is
defined
to
be
the
output
data
of
the
second
algorithm
of
(∗
×
)
for
some
topological
group
isomorphic
to
G
k
;
an
isomorphism
of
collections
of
coric
data
is
defined
to
be
a
Γ-multiple
of
the
isomorphism
between
collections
of
output
data
of
(∗
×
)
associated
to
an
isomorphism
of
topological
groups.
The
radial
algorithm
is
the
algorithm
given
by
the
assignment
×
(Π),
G
O
×
(G),
α
×
)
→
(G
O
×
(G))
(Π
M
TM
—
whose
associated
radial
functor
is
full
and
essentially
surjective,
hence
determines
a
multiradial
environment.
(iv)
By
considering
the
subgroups
of
torsion
elements
of
the
various
ind-topo-
logical
monoids
that
appeared
in
(ii)
and
(iii),
one
obtains
functorial
group-theoretic
algorithms
Π
→
μ
(Π));
(Π
M
TM
G
→
(G
O
μ
(G))
(∗
μ
)
defined,
respectively,
on
topological
groups
Π
isomorphic
to
Π
tp
X
and
G
isomor-
k
phic
to
G
k
—
i.e.,
a
“cyclotomic
version”
of
the
algorithms
of
(∗
×
)
[cf.
(iii)].
×μ
μ
×
(−)
=
M
TM
(−)/M
TM
(−),
O
×μ
(−)
=
Moreover,
by
forming
the
quotients
M
TM
×
μ
O
(−)/O
(−),
one
obtains
functorial
group-theoretic
algorithms
def
Π
→
×μ
(Π
M
TM
(Π));
G
→
def
(G
O
×μ
(G))
(∗
×μ
)
defined,
respectively,
on
topological
groups
Π
isomorphic
to
Π
tp
X
and
G
isomorphic
k
to
G
k
—
i.e.,
a
“co-cyclotomic
version”
of
the
algorithms
of
(∗
×
)
[cf.
(iii)].
Now
one
verifies
easily
that
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
39
by
replacing
the
symbol
“×”
in
(iii)
by
the
symbol
“μ”
or,
alternatively,
by
the
symbol
“×μ”,
one
obtains,
respectively,
“cyclotomic”
and
“co-cyclotomic”
versions
of
the
example
treated
in
(iii).
In
the
case
of
“×μ”,
let
us
write
Ism(G)
for
the
compact
topological
group
of
G-isometries
of
O
×μ
(G),
i.e.,
G-equivariant
automorphisms
of
the
ind-topological
module
O
×μ
(G)
that,
for
each
open
subgroup
H
⊆
G,
preserve
the
“lattice”
in
O
×μ
(G)
H
determined
by
the
image
of
O
×
(G)
H
[i.e.,
where
the
superscript
“H”
denotes
the
submodule
of
H-invariants].
Let
Γ
×μ
⊆
Ism(−)
be
a
closed
subgroup,
i.e.,
a
collection
of
closed
subgroups
of
each
Ism(G)
that
is
∼
preserved
by
arbitrary
isomorphisms
of
topological
groups
G
1
→
G
2
.
Then
one
verifies
easily
that,
in
the
“co-cyclotomic”
version
discussed
above
of
the
example
treated
in
(iii),
one
may
replace
the
“Γ”
in
(iii)
by
such
a
“Γ
×μ
”.
Finally,
we
observe
that
one
example
of
such
a
“Γ
×μ
”
—
which
we
shall
denote
by
means
of
the
notation
Ism
—
is
the
case
where
one
takes
Γ
×μ
to
be
the
entire
group
“Ism(−)”;
another
example
of
such
a
“Γ
×μ
”
is
the
image
Im(
Z
×
)
of
the
natural
homomorphism
Z
×
Z
×
p
→
Ism.
(v)
Another
example
of
a
radial
environment
may
be
obtained
as
follows.
We
define
a
collection
of
radial
data
μ
(Π),
G
O
×μ
(G),
α
μ,×μ
)
(Π
M
TM
to
consist
of
the
output
data
of
the
first
algorithm
of
(∗
μ
)
associated
to
a
topological
group
Π
isomorphic
to
Π
tp
X
,
the
output
data
of
the
second
algorithm
of
(∗
×μ
)
k
associated
to
a
topological
group
G
isomorphic
to
G
k
,
and
the
poly-morphism
[cf.
[IUTchI],
§0]
of
ind-topological
modules
equipped
with
topological
group
actions
μ
α
μ,×μ
:
(Π
M
TM
(Π))
→
(G
O
×μ
(G))|
Π
∼
determined
by
the
full
poly-isomorphism
Π/Δ
→
G
[cf.
(i)]
and
the
trivial
ho-
μ
(Π)
→
O
×μ
(G)
—
i.e.,
the
composite
of
the
natural
homomor-
momorphism
M
TM
∼
∼
μ
×
phisms
M
TM
(Π)
⊆
M
TM
(Π)
→
O
×
(G)
O
×μ
(G)
[where
the
“
→
”
arises
from
the
poly-isomorphism
α
×
of
(iii)].
An
isomorphism
of
collections
of
radial
data
∼
μ
μ
∗
(Π
M
TM
(Π),
G
O
×μ
(G),
α
μ,×μ
)
→
(Π
∗
M
TM
(Π
∗
),
G
∗
O
×μ
(G
∗
),
α
μ,×μ
)
is
defined
to
consist
of
the
isomorphism
of
ind-topological
modules
equipped
with
∼
μ
μ
(Π))
→
(Π
∗
M
TM
(Π
∗
))
induced
by
an
topological
group
actions
(Π
M
TM
∼
isomorphism
of
topological
groups
Π
→
Π
∗
,
together
with
a
Γ
×μ
-multiple
of
the
isomorphism
of
ind-topological
modules
equipped
with
topological
group
actions
40
SHINICHI
MOCHIZUKI
∼
(G
O
×μ
(G))
→
(G
∗
O
×μ
(G
∗
))
induced
by
an
isomorphism
of
topological
∼
groups
G
→
G
∗
[so
one
verifies
immediately
that
these
isomorphisms
are
compat-
∗
in
the
evident
sense].
A
collection
of
coric
data
is
defined
ible
with
α
μ,×μ
,
α
μ,×μ
to
be
the
output
data
of
the
second
algorithm
of
(∗
×μ
)
for
some
topological
group
isomorphic
to
G
k
;
an
isomorphism
of
collections
of
coric
data
is
defined
to
be
a
Γ
×μ
-multiple
of
the
isomorphism
between
collections
of
output
data
of
(∗
×μ
)
asso-
ciated
to
an
isomorphism
of
topological
groups.
[That
is
to
say,
the
definition
of
the
coric
data
is
the
same
as
in
the
“co-cyclotomic”
version
discussed
in
(iv).]
The
radial
algorithm
is
the
algorithm
given
by
the
assignment
μ
(Π),
G
O
×μ
(G),
α
μ,×μ
)
→
(G
O
×μ
(G))
(Π
M
TM
—
whose
associated
radial
functor
is
full
and
essentially
surjective,
hence
determines
a
multiradial
environment.
μ
(Π)”
in
the
discussion
of
(v)
by
the
no-
(vi)
By
replacing
the
notation
“M
TM
Θ
tation
“Π
μ
(M
∗
(Π))
⊗
Q/Z”
[cf.
Propositions
1.2,
(i);
1.5,
(i),
(iii)],
one
verifies
immediately
that
one
obtains
an
“exterior-cyclotomic
version”
of
the
multiradial
environment
constructed
in
(v).
(vii)
In
the
discussion
to
follow,
we
shall
also
consider
the
functorial
group-
theoretic
algorithms
Π
→
gp
(Π));
(Π
M
TM
G
→
(G
O
gp
(G))
(∗
gp
)
obtained
by
passing
to
the
respective
groupifications
of
the
monoids
M
TM
(Π),
O
(G),
as
well
as
the
functorial
group-theoretic
algorithms
Π
→
gp
(Π
M
TM
(Π));
G
→
(G
O
gp
(G))
(∗
gp
)
obtained
by
passing
to
the
respective
inductive
limits
of
the
profinite
completions
gp
(Π)
J
,
O
gp
(G)
J
[i.e.,
where
the
superscript
“J”
denotes
the
submodule
of
J-
of
M
TM
invariants],
as
J
ranges
over
the
open
subgroups
of
Π
or
G.
Thus,
there
is
a
natural
gp
(Π),
O
gp
(G);
by
action
of
Γ
on
the
underlying
ind-topological
modules
of
M
TM
considering
the
Γ-orbit
of
the
poly-isomorphism
induced
by
the
poly-isomorphism
α
of
(ii),
one
obtains
a
poly-isomorphism
gp
α
gp
:
(Π
M
TM
(Π))
∼
→
(G
O
gp
(G))|
Π
that
is
compatible
[in
the
evident
sense]
with
the
poly-isomorphism
α
×
of
(iii).
(viii)
The
following
example
of
a
radial
environment
is
another
variant
of
the
example
of
(iii).
We
define
a
collection
of
radial
data
(Π
M
TM
(Π),
G
O
gp
(G),
α
,×μ
)
to
consist
of
the
output
data
of
the
algorithm
of
(∗
TM
)
associated
to
a
topologi-
cal
group
Π
isomorphic
to
Π
tp
X
,
the
output
data
of
the
second
algorithm
of
(∗
gp
)
k
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
41
[cf.
(vii)]
associated
to
a
topological
group
G
isomorphic
to
G
k
,
and
the
follow-
ing
diagram
α
,×μ
of
poly-morphisms
of
ind-topological
monoids
equipped
with
topological
group
actions
(Π
M
TM
(Π))
→
gp
(Π
M
TM
(Π))
∼
(G
O
gp
(G))|
Π
→
←
(G
O
×
(G))|
Π
(G
O
×μ
(G))|
Π
∼
—
where
the
“
→
”
denotes
the
natural
inclusion;
the
“
→
”
denotes
the
poly-
isomorphism
α
gp
of
(vii);
the
“
←
”
denotes
the
natural
inclusion;
the
“
”
denotes
the
natural
surjection.
An
isomorphism
of
collections
of
radial
data
(Π
∼
∗
M
TM
(Π),
G
O
gp
(G),
α
,×μ
)
→
(Π
∗
M
TM
(Π
∗
),
G
∗
O
gp
(G
∗
),
α
,×μ
)
is
defined
to
consist
of
the
isomorphism
of
ind-topological
monoids
equipped
with
∼
topological
group
actions
(Π
M
TM
(Π))
→
(Π
∗
M
TM
(Π
∗
))
induced
by
an
∼
isomorphism
of
topological
groups
Π
→
Π
∗
,
together
with
a
Γ-multiple
of
the
isomorphism
of
ind-topological
modules
equipped
with
topological
group
actions
∼
(G
O
gp
(G))
→
(G
∗
O
gp
(G
∗
))
induced
by
an
isomorphism
of
topological
∼
groups
G
→
G
∗
[so
one
verifies
immediately
that
these
isomorphisms
are
compatible
∗
in
the
evident
sense];
here,
we
note
that
any
such
isomorphism
with
α
,×μ
,
α
,×μ
∼
∼
(G
O
gp
(G))
→
(G
∗
O
gp
(G
∗
))
induces
isomorphisms
(G
O
×
(G))
→
(G
∗
∼
O
×
(G
∗
)),
(G
O
×μ
(G))
→
(G
∗
O
×μ
(G
∗
))
in
a
fashion
compatible
with
α
,×μ
,
∗
.
The
definition
of
coric
data
and
isomorphisms
of
collections
of
coric
data
is
α
,×μ
the
same
as
in
(v)
[i.e.,
where
one
takes
“Γ
×μ
”
to
be
the
image
Im(Γ)
of
Γ
⊆
Z
×
].
The
radial
algorithm
is
the
algorithm
given
by
the
assignment
(Π
M
TM
(Π),
G
O
gp
(G),
α
,×μ
)
→
(G
O
×μ
(G))
—
whose
associated
radial
functor
is
full
and
essentially
surjective,
hence
determines
a
multiradial
environment.
(ix)
Note
that
if
G
is
a
topological
group
isomorphic
to
G
k
,
then,
in
addi-
tion
to
G
O
×
(G),
G
O
×μ
(G),
one
may
also
construct
the
log-shell
I(G)
⊆
O
×μ
(G)
[i.e.,
p
−1
times
the
image
of
the
G-invariants
of
O
×
(G)
in
O
×μ
(G)
—
cf.
[AbsTopIII],
Proposition
5.8,
(ii)].
In
particular,
if
one
replaces
the
nota-
tion
“G
O
×μ
(G)”
in
the
discussion
of
(v),
(vi),
and
(viii)
by
the
notation
“(G
O
×μ
(G),
I(G)
⊆
O
×μ
(G))”
[i.e.,
“G
O
×μ
(G)
equipped
with
its
associ-
ated
log-shell”],
then
one
verifies
immediately
that
one
obtains
a
“log-shell
version”
of
the
multiradial
environments
constructed
in
(v),
(vi),
and
(viii).
Remark
1.8.1.
In
the
context
of
the
various
examples
given
in
Example
1.8,
(iii),
(iv),
(v),
(vi),
(vii),
(viii),
and
(ix),
it
is
useful
to
note
that
no
automorphism
of
O
×μ
(G)
induced
by
an
element
of
Aut(G)
[e.g.,
an
element
of
G,
regarded
as
an
inner
automorphism
of
G]
coincides
with
an
automorphism
of
O
×μ
(G)
induced
by
an
element
of
Γ
that
has
nontrivial
image
in
Z
×
p
.
42
SHINICHI
MOCHIZUKI
Indeed,
this
follows
immediately
by
observing
that
the
composite
with
the
p-adic
logarithm
of
the
cyclotomic
character
of
G
determines
[in
light
of
the
definition
of
O
×
(G),
in
terms
of
abelianizations
of
open
subgroups
of
G
—
cf.
[AbsTopIII],
Proposition
5.8,
(i)]
a
natural
surjection
O
×μ
(G)
Q
p
,
which
[cf.,
e.g.,
[AbsAnab],
Proposition
1.2.1,
(vi)]
is
Aut(G)-equivariant,
relative
to
the
trivial
action
of
Aut(G)
on
Q
p
,
and
Γ-equivariant,
relative
to
the
natural
action
of
Γ
⊆
Z
×
[via
the
natural
surjection
Z
×
Z
×
p
]
on
Q
p
.
Example
1.9.
Radial
and
Coric
Data
III:
Graphs
of
Functorial
Group-
theoretic
Algorithms.
(i)
Let
E
and
F
be
categories
that
arise
from
“types
of
mathematical
data”
[cf.
the
discussion
of
Example
1.7,
(i)];
Ξ
:
E
→
F
a
functor
that
arises
from
a
“functorial
algorithm”
[cf.
the
discussion
of
Example
1.7,
(i)].
Then
one
may
define
a
new
category
G
—
that
also
arises
from
a
“type
of
mathematical
data”
—
as
follows:
the
objects
of
G
are
pairs
(E,
Ξ(E)),
where
E
∈
Ob(E),
and
Ξ(E)
∈
Ob(F)
is
the
image
of
E
via
Ξ;
the
morphisms
of
G
are
the
pairs
of
arrows
(f
:
E
→
E
,
Ξ(f
)
:
Ξ(E)
→
Ξ(E
)).
We
shall
refer
to
G
[or
the
“type
of
mathematical
data”
that
gives
rise
to
G]
as
the
graph
of
Ξ.
Note
that
this
construction
was
applied,
in
effect,
in
the
discussion
of
the
various
radial
environments
constructed
in
Example
1.8.
Finally,
we
observe
that
we
have
natural
functors
E
→
G
[given
by
E
→
(E,
Ξ(E))],
G
→
E
[given
by
(E,
Ξ(E))
→
E],
G
→
F
[given
by
(E,
Ξ(E))
→
Ξ(E)].
(ii)
In
the
notation
of
(i),
suppose
that
E
is
the
category
of
topological
groups
isomorphic
to
Π
tp
X
and
isomorphisms
of
topological
groups,
and
that
Ξ
is
some
k
“functorial
group-theoretic
algorithm”
[whose
input
data
consists
of
a
topological
group
isomorphic
to
Π
tp
X
].
Let
(R,
C,
Φ)
be
the
radial
environment
of
Example
1.8,
k
(i).
Then
composing
the
functor
R
→
E
given
by
the
assignment
(Π,
G,
α)
→
Π
with
Ξ
:
E
→
F
yields
a
functor
R
→
F,
whose
graph
we
denote
by
R
†
.
Thus,
by
considering
the
natural
functors
Ψ
R
:
R
→
R
†
[cf.
(i)],
R
†
→
R
→
C,
and
taking
def
C
†
=
C,
we
obtain
a
diagram
as
in
the
display
of
Example
1.7,
(iv).
Since
(R,
C,
Φ)
is
a
multiradial
environment,
it
thus
follows
that
Ψ
R
is
multiradially
defined
[cf.
Example
1.7,
(iv)].
That
is
to
say,
by
using
the
radial
environment
of
Example
1.8,
(i),
one
concludes
that
any
“functorial
group-theoretic
algorithm”
whose
input
data
consists
of
a
topological
group
isomorphic
to
Π
tp
X
gives
rise
—
in
a
tautological
fashion
k
[cf.
the
discussion
of
tautological
multiradializations
in
Example
1.7,
(ii)]
—
to
a
multiradially
defined
functor.
This
approach
will
be
discussed
further
in
Remark
1.9.1
below.
(iii)
On
the
other
hand,
one
may
also
construct
a
radial
environment
as
follows.
We
define
a
collection
of
radial
data
to
be
a
topological
group
Π
isomorphic
to
Π
tp
X
,
k
and
an
isomorphism
of
collections
of
radial
data
to
be
an
isomorphism
of
topological
groups.
The
definitions
of
coric
data
and
isomorphisms
of
collections
of
coric
data
are
the
same
as
in
Example
1.8,
(i).
The
radial
functor
Φ
:
R
→
C
is
defined
via
the
assignment
Π
→
Π/Δ
[cf.
the
notation
of
Example
1.8,
(i)].
Thus,
Φ
fails
to
be
full
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
43
[cf.,
e.g.,
[AbsTopIII],
§I3;
[AbsTopIII],
Remark
1.9.1].
That
is
to
say,
(R,
C,
Φ)
is
a
uniradial
environment.
Now
suppose
that
Ξ
:
E
→
F
is
as
in
(ii).
Then
since
R
may
be
identified
with
E,
the
graph
of
Ξ
:
R
=
E
→
F
yields
a
category
R
†
equipped
def
with
natural
functors
Ψ
R
:
R
→
R
†
,
Φ
†
:
R
†
→
R
→
C
†
=
C.
In
particular,
we
obtain
a
diagram
as
in
the
display
of
Example
1.7,
(iv).
Since
(R,
C,
Φ)
is
a
uniradial
environment,
it
thus
follows
that
Ψ
R
is
uniradially
defined
[cf.
Example
1.7,
(iv)].
That
is
to
say,
by
using
the
radial
environment
just
defined,
one
concludes
that
any
“functorial
group-theoretic
algorithm”
whose
input
data
consists
of
a
topological
group
isomorphic
to
Π
tp
X
also
gives
rise
—
in
a
tautological
k
fashion
—
to
a
uniradially
defined
functor.
This
approach
will
be
discussed
further
in
Remark
1.9.1
below.
(iv)
Let
Π
be
a
topological
group
isomorphic
to
Π
tp
X
;
Δ
⊆
Π
the
subgroup
∼
k
of
Example
1.8,
(i).
Recall
the
isomorphism
“μ
Z
(G
k
)
→
μ
Z
(Π
X
)”
of
[AbsTopIII],
Corollary
1.10,
(c),
which
is
constructed
by
means
of
a
“functorial
group-theoretic
algorithm”.
The
inverse
of
this
isomorphism
yields
a
cyclotomic
rigidity
isomor-
phism
∼
(l
·
Δ
Θ
)(Π)
→
μ
Z
(Π/Δ)
[cf.
the
discussion
of
Proposition
1.3,
(ii)]
—
where
we
write
“(l
·
Δ
Θ
)(Π)”
for
the
[group-theoretic!]
subquotient
of
Π
discussed
in
[EtTh],
Corollary
2.18,
(i).
Thus,
in
summary,
one
has
a
“functorial
group-theoretic
algorithm”
whose
input
data
consists
of
the
topological
group
Π,
and
whose
output
data
may
be
thought
of
as
consisting
of
Π,
the
two
topological
Π-modules
“(l
·
Δ
Θ
)(Π)”,
“μ
Z
(Π/Δ)”,
and
∼
the
above
isomorphism
of
Π-modules
(l
·
Δ
Θ
)(Π)
→
μ
Z
(Π/Δ).
Thus,
if
one
takes
this
“functorial
group-theoretic
algorithm”
to
be
the
algorithm
that
gives
rise
to
the
functor
Ξ
in
the
discussion
of
(ii)
and
(iii),
then
one
concludes
that
the
above
∼
cyclotomic
rigidity
isomorphism
(l
·
Δ
Θ
)(Π)
→
μ
Z
(Π/Δ)
may
be
thought
of
as
giving
rise
to
either
(a)
a
multiradially
defined
functor,
via
the
approach
of
(ii),
or
(b)
a
uniradially
defined
functor,
via
the
approach
of
(iii).
On
the
other
hand,
there
is
also
another
way
to
obtain
a
multiradially
defined
functor
from
this
cyclotomic
rigidity
isomorphism,
as
follows.
Let
(R,
C,
Φ)
be
the
multiradial
environment
of
Example
1.8,
(i).
Now
define
a
collection
of
daggered
radial
data
∼
(Π,
G,
α,
(l
·
Δ
Θ
)(Π)
→
μ
Z
(G))
to
consist
of
radial
data
(Π,
G,
α)
as
in
Example
1.8,
(i),
together
with
the
poly-
∼
isomorphism
(l
·
Δ
Θ
)(Π)
→
μ
Z
(G)
obtained
by
composing
the
above
cyclotomic
∼
rigidity
isomorphism
“(l·Δ
Θ
)(Π)
→
μ
Z
(Π/Δ)”
with
the
poly-isomorphism
μ
Z
(Π/Δ)
∼
∼
→
μ
Z
(G)
induced
by
the
poly-isomorphism
α
:
Π/Δ
→
G.
Thus,
the
poly-
∼
isomorphism
(l
·
Δ
Θ
)(Π)
→
μ
Z
(G)
consists
not
of
a
single
isomorphism
of
topo-
logical
modules,
but
rather
of
an
Aut(G)-orbit
—
or,
more
precisely,
a
Γ-orbit,
where
Γ
⊆
Z
×
is
the
image
of
Aut(G)
via
the
cyclotomic
character
on
Aut(G)
[cf.
[AbsAnab],
Proposition
1.2.1,
(vi)]
—
of
isomorphisms
of
topological
modules.
An
44
SHINICHI
MOCHIZUKI
isomorphism
of
collections
of
daggered
radial
data
is
defined
to
be
an
isomorphism
between
the
underlying
collections
of
radial
data
[which
is
necessarily
compatible
with
the
poly-isomorphism
of
topological
modules
that
constitutes
the
final
member
def
of
the
collections
of
daggered
radial
data
in
question].
Thus,
if
we
take
C
†
=
C,
then
the
“functorial
group-theoretic
algorithm”
that
gives
rise
to
the
cyclotomic
rigid-
∼
ity
isomorphism
“(l
·
Δ
Θ
)(Π)
→
μ
Z
(Π/Δ)”
yields
a
functor
Ψ
R
:
R
→
R
†
[that
arises
from
a
“functorial
algorithm”],
together
with
a
diagram
as
in
the
display
of
Example
1.7,
(iv).
That
is
to
say,
(c)
this
multiradially
defined
functor
Ψ
R
:
R
→
R
†
yields
an
alternative
[i.e.,
relative
to
(a)]
multiradial
approach
to
representing
the
“functorial
group-
theoretic
algorithm”
that
gives
rise
to
the
cyclotomic
rigidity
isomorphism
∼
“(l
·
Δ
Θ
)(Π)
→
μ
Z
(Π/Δ)”.
This
is
the
approach
taken
in
Corollary
1.11,
(b),
below.
Remark
1.9.1.
In
general,
the
portion
of
the
“functorial
group-theoretic
algo-
rithm”
that
appears
in
the
discussion
of
Example
1.9,
(ii),
(iii),
and
(iv),
which
involves
the
quotient
Π/Δ
of
Π
will
depend
not
only
on
the
structure
of
the
ab-
stract
topological
group
underlying
Π/Δ,
but
also
on
the
structure
of
Π/Δ
as
a
quotient
of
Π
—
i.e.,
from
the
point
of
view
of
the
discussion
of
Example
1.8,
(i),
on
the
“arithmetic
holomorphic
structure”
on
the
topological
group
Π/Δ
determined
by
this
quotient
structure.
In
fact,
the
original
motivation
for
the
introduction
of
the
“multiradial
terminology”
of
Example
1.7
was
precisely
to
study
the
extent
to
which
such
“functorial
group-theoretic
algorithms”
could
be
formulated
in
such
a
way
as
to
compute
which
portions
of
the
output
data
of
such
algorithms
do
indeed
depend
in
an
essential
way
on
the
“arithmetic
holomorphic
structure”
and
which
portions
are
“mono-analytic”
[cf.
[AbsTopIII],
§I3],
i.e.,
depend
only
on
the
structure
of
the
topological
group
Π/Δ
[which
one
thinks
of
as
a
sort
of
“underlying
arithmetic
real
analytic
structure”
of
the
“arithmetic
holomorphic
structures”
involved].
From
this
point
of
view,
the
tautological
approach
of
Example
1.9,
(ii)
[i.e.,
Example
1.9,
(iv),
(a)],
may
be
thought
of
as
expressing
the
idea
that
if
one
thinks
of
each
of
the
quotients
“Π/Δ”
in
the
“spokes”
of
Fig.
1.2
as
being
equipped
with
a
fixed
“arithmetic
holomorphic
structure”
and
hence
only
related
to
the
coric
“G”
via
some
indeterminate
isomorphism
of
topological
groups,
then
one
obtains
a
multiradially
defined
functor,
i.e.,
a
functor
that
is
tautologically
compatible
with
mono-analytic
deformations
of
the
various
“arithmetic
holomorphic
structures”
that
one
might
impose
on
the
coric
“G”.
Put
another
way,
this
multiradially
defined
algorithm
is
an
algorithm
that
is
tautologically
compatible
with
simultaneous
execution
on
multiple
spokes
of
Fig.
1.2.
By
contrast,
the
tautological
approach
of
Example
1.9,
(iii)
[i.e.,
Example
1.9,
(iv),
(b)],
may
be
thought
of
as
expressing
the
idea
that
if
one
tries
to
identify
the
various
quotients
“Π/Δ”
in
the
“spokes”
of
Fig.
1.2
via
arbitrary
mono-analytic
isomorphisms,
then
one
only
obtains
a
uniradially
defined
functor,
i.e.,
a
functor
that
fails
to
be
compatible
with
mono-analytic
identifications
[i.e.,
gluing
isomorphisms]
of
the
various
“arithmetic
holomorphic
structures”
on
the
coric
“G”.
Put
another
way,
this
uniradially
defined
algorithm
is
an
algorithm
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
45
that
can
only
be
consistently
executed
on
one
spoke
at
a
time.
Finally,
the
ap-
proach
of
Example
1.9,
(iv),
(c),
expresses
the
idea
that,
in
the
case
of
the
particular
cyclotomic
rigidity
isomorphism
under
consideration,
if
one
weakens
the
rigidity
of
this
isomorphism
by
working
with
this
isomorphism
up
to
a
certain
indeterminacy,
then
one
may
construct
a
multiradially
defined
functor,
i.e.,
a
functor
that
is
indeed
compatible
with
mono-analytic
identifications
[i.e.,
gluing
isomorphisms]
of
the
var-
ious
“arithmetic
holomorphic
structures”
on
the
coric
“G”,
albeit
up
to
a
certain
specified
indeterminacy.
Thus,
the
multiradiality
obtained
in
Example
1.9,
(iv),
(c),
depends,
in
an
essential
way,
on
the
content
of
the
“functorial
group-theoretic
algorithm”
involved.
This
approach
taken
in
Example
1.9,
(iv),
(c),
is
representa-
tive
of
the
approach
taken
in
Corollaries
1.10,
1.11,
and
1.12
below,
which
may
be
thought
of
as
“computations”
of
the
“certain
indeterminacy”
that
one
must
allow
in
order
to
construct
a
multiradially
defined
functor
as
in
Example
1.9,
(iv),
(c).
Remark
1.9.2.
(i)
One
way
to
summarize
the
discussion
of
Remark
1.9.1
is
as
follows.
If
uniradially
defined
functors
correspond
to
constructions
that
depend,
in
a
strict
sense,
on
the
“arithmetic
holomorphic
structure”,
while
corically
defined
functors
correspond
to
constructions
that
only
depend
on
the
underlying
mono-analytic
structure
[i.e.,
“arithmetic
real
analytic
structure”],
then
multiradially
defined
functors
correspond
to
constructions
that
depend
on
the
“arithmetic
holomorphic
structure”,
but
only
in
a
fashion
that
is
compatible
with
a
given
description
of
how
this
arithmetic
holomorphic
structure
is
related
to
—
e.g.,
“embedded
in”
—
the
underlying
mono-
analytic
structure.
For
instance,
in
the
various
multiradial
environments
of
Example
1.8,
this
descrip-
tion
of
the
relation
to
the
underlying
mono-analytic
structure
is
given,
at
a
concrete
level,
by
the
various
poly-morphisms
[or
diagrams
of
poly-morphisms]
“α
(−)
”
that
appear
in
the
radial
data
of
these
multiradial
environments.
This
point
of
view
is
summarized
in
Fig.
1.3
below.
(ii)
From
the
point
of
view
of
the
analogy
with
connections
discussed
in
Remark
1.7.1,
one
may
think
of
a
multiradial
environment
as
a
structure
that
allows
one
to
execute
parallel
transport
operations
between
distinct
arithmetic
holomorphic
stuctures,
i.e.,
to
describe
what
one
arithmetic
holomorphic
structure
looks
like
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
that
is
only
related
to
the
original
arithmetic
holomorphic
structure
via
the
mono-analytic
core.
(iii)
From
the
point
of
view
of
the
analogy
with
connections
discussed
in
Re-
mark
1.7.1,
it
is
also
interesting
to
observe
that
one
may
think
of
the
different
ap-
proaches
to
multiradiality
discussed
in
Example
1.9,
(iv),
(a),
(c),
as
being
roughly
analogous
to
the
phenomenon
of
distinct
connection
structures
on
a
single
fibration.
Moreover,
of
these
different
approaches,
the
tautological,
“general
non-
sense”
approach
of
Example
1.9,
(iv),
(a),
is,
in
some
sense,
[not
surprisingly!]
the
“least
interesting”
[although
it
will
at
times
be
of
use
in
the
theory
of
the
present
series
of
papers!].
This
sort
of
“general
nonsense”
approach
is
reminiscent
of
the
46
SHINICHI
MOCHIZUKI
tautological
approach
to
constructing
connections
that
occurs
in
the
p-adic
theory
of
the
crystalline
site,
i.e.,
by
simply
forming
the
tensor
product
with
the
ring
of
functions
of
the
PD-envelope
along
the
diagonal
of
the
fiber
product
of
two
copies
of
the
space
under
consideration.
From
the
point
of
view
of
the
issue
of
“describing
what
one
arithmetic
holomorphic
structure
looks
like
from
the
point
of
view
of
another”
[cf.
(ii)],
the
“tautological”
approach
is
not
very
interesting
precisely
because
it
involves
working,
in
effect,
with
the
“tautological”
collection
of
“labels
of
all
possible
arithmetic
holo-
morphic
structures”
—
i.e.,
corresponding
to
the
various
choices
of
one
particular
arrow
among
the
arrows
that
constitute
the
poly-morphism
denoted
“α”
in
Example
1.8,
(i)
—
without
describing
in
further,
more
explicit
terms
what
these
various
“alien”
arithmetic
holomorphic
struc-
tures
look
like
relative
to
structures
determined
by
a
given
arithmetic
holomorphic
structure.
By
contrast,
the
“non-tautological”
approach
to
multiradiality
of
Example
1.9,
(iv),
(c),
by
means
of
the
explicit
computation
of
indeterminacies
is
much
more
interesting
in
that
it
yields
a
more
detailed,
explicit
description
of
a
structure
[e.g.,
a
cyclotomic
rigidity
isomorphism]
associated
to
an
“alien”
arithmetic
holomorphic
structure
in
terms
of
the
structure
associated
to
a
given
arithmetic
holomorphic
structure.
abstract
general
nonsense
inter-universal
Teichmüller
theory
classical
complex
Teichmüller
theory
uniradially
defined
functors
arithmetic
holomorphic
structures
holomorphic
structures
multiradially
defined
functors
arithmetic
holomorphic
structures
described
in
terms
of
underlying
mono-analytic
structures
holomorphic
structures
described
in
terms
of
underlying
real
analytic
structures
corically
defined
functors
underlying
mono-analytic
structures
underlying
real
analytic
structures
Fig.
1.3:
Uniradiality,
Multiradiality,
and
Coricity
We
now
proceed
to
discuss
our
main
results
concerning
multiradiality.
Corollary
1.10.
(Multiradial
Mono-theta
Cyclotomic
Rigidity
Isomor-
phisms)
Write
(R,
C,
Φ
:
R
→
C)
—
i.e.,
in
the
notation
of
Example
1.8,
(v),
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
47
(vi),
×μ
(Π
Π
μ
(M
Θ
(G),
α
μ,×μ
)
→
(G
O
×μ
(G))
∗
(Π))
⊗
Q/Z,
G
O
—
for
the
multiradial
environment
constituted
by
the
exterior-cyclotomic
version
[cf.
Example
1.8,
(vi)]
of
the
multiradial
environment
discussed
in
Example
1.8,
(v).
Consider
the
cyclotomic
rigidity
isomorphism
∼
(l
·
Δ
Θ
)(Π)
→
Π
μ
(M
Θ
∗
(Π))
(∗
mono-Θ
)
Π
[where
we
identify
(l
·Δ
Θ
)(M
Θ
∗
(Π))
with
(l
·Δ
Θ
)(Π)
—
cf.
Proposition
1.4]
obtained
by
composing
the
functorial
algorithm
Π
→
M
Θ
∗
(Π)
of
Proposition
1.2,
(i)
[cf.
also
Proposition
1.5,
(i)],
with
the
functorial
algorithm
for
constructing
a
cyclotomic
rigidity
isomorphism
of
Proposition
1.5,
(iii).
Then
the
data
consisting
of
the
topo-
logical
group
Π,
the
topological
Π-modules
constituted
by
the
domain
and
codomain
),
and
the
isomorphism
(∗
mono-Θ
)
determines
a
functor
R
→
F
[i.e.,
of
(∗
mono-Θ
Π
Π
where
F
denotes
the
category
defined
in
the
evident
way
so
as
to
accommodate
the
data
just
listed]
which
arises
from
a
functorial
algorithm
in
the
topological
group
Π;
denote
the
corresponding
graph
[cf.
Example
1.9,
(i)]
by
R
†
.
In
particular,
the
resulting
natural
functor
Ψ
R
:
R
→
R
†
[cf.
Example
1.9,
(i)]
is
multiradially
defined.
Proof.
The
various
assertions
of
Corollary
1.10
follow
immediately
from
the
defi-
nitions
involved.
Remark
1.10.1.
We
recall
in
passing
that
the
domain
and
codomain
of
the
iso-
)
of
Corollary
1.10,
as
well
as
the
isomorphism
(∗
mono-Θ
)
itself,
morphism
(∗
mono-Θ
Π
Π
are
constructed
from
various
subquotients
of
[the
projective
system
of
topological
which
are
completely
determined
by
the
structure
of
Δ
M
Θ
as
groups]
Δ
M
Θ
∗
(Π)
∗
(Π)
determined
by
a
projective
system
of
topological
groups,
the
subgroups
of
Δ
M
Θ
∗
(Π)
the
images
of
the
“theta
section”
portions
of
the
system
of
mono-theta
environ-
ments
M
Θ
∗
(Π),
and
the
images
[arising
from
the
natural
outer
actions
involved
Θ
).
In-
—
cf.
Definition
1.1,
(i)]
of
(l
·
Z)(M
Θ
∗
(Π))
and
G(M
∗
(Π))
in
Out(Δ
M
Θ
∗
(Π)
deed,
the
algorithms
described
in
the
proofs
of
[EtTh],
Corollary
2.18,
(i),
(iii);
[EtTh],
Corollary
2.19,
(i),
for
constructing
the
various
subquotients
of
Δ
M
Θ
∗
(Π)
corresponding
to
the
domain
and
codomain
of
(∗
mono-Θ
),
as
well
as
to
the
graph
Π
)
itself,
depend
only
on
the
structure
of
the
projective
of
the
isomorphism
(∗
mono-Θ
Π
Θ
system
of
topological
groups
Δ
M
∗
(Π)
[cf.,
e.g.,
[EtTh],
Proposition
2.11,
(i)],
the
determined
by
the
images
of
the
“theta
section”
portions
of
subgroups
of
Δ
M
Θ
∗
(Π)
the
system
of
mono-theta
environments
M
Θ
∗
(Π)
[cf.
[EtTh],
Definition
2.13,
(ii),
(c)],
and
the
construction
of
the
group
Δ
C
(Π)
[which
was
reviewed
in
Proposition
Θ
∼
)
Δ
Y
(M
Θ
1.6,
(i)]
containing
(Δ
M
Θ
∗
(Π))
⊆
Δ
X
(M
∗
(Π))
=
Δ,
which
is
used
to
∗
(Π)
construct
the
various
subquotients
that
appear
in
the
crucial
[EtTh],
Proposition
2.12;
[EtTh],
Proposition
2.14,
(i).
Remark
1.10.2.
follows:
In
words,
the
content
of
Corollary
1.10
may
be
understood
as
48
SHINICHI
MOCHIZUKI
Since
O
×μ
(G)
is
constructed
by
forming
the
quotient
of
O
×
(G)
by
the
roots
of
unity
[i.e.,
O
μ
(G)]
—
recall
the
triviality
of
the
homomorphism
×μ
(G)
[cf.
Example
1.8,
(v),
(vi)]!
—
any
rigidifi-
Π
μ
(M
Θ
∗
(Π))⊗Q/Z
→
O
cation
of
the
cyclotome
Π
μ
(M
Θ
∗
(Π))
that
depends
only
on
the
structure
of
the
mono-theta-environment
M
Θ
∗
(Π)
will
be
tautologically
com-
patible
with
the
coricity
of
O
×μ
(G),
i.e.,
with
the
“sharing
of
O
×μ
(G)”
by
distinct
arithmetic
holomorphic
structures
[cf.
the
discussion
of
Remark
1.9.1;
Fig.
1.4
below].
This
contrasts
sharply
with
the
situation
to
be
considered
in
Corollary
1.11
below
—
cf.
Remarks
1.11.3,
1.11.4,
below.
A
similar
statement
may
be
made
concerning
∼
the
subquotient
(l
·
Δ
Θ
)(Π)
of
Δ
⊆
Π,
which
maps
trivially
to
Π/Δ
→
G.
i
i
Π
...
Π
⏐
⏐
...
−→
G
O
×μ
(G)
Γ
×μ
←−
...
⏐
⏐
...
i
i
Π
Π
Fig.
1.4:
A
single
coric
pair
G
O
×μ
(G),
regarded
up
to
the
action
of
Γ
×μ
Remark
1.10.3.
In
the
context
of
Corollary
1.10,
it
is
useful
to
recall
the
following
[cf.
the
discussion
of
[EtTh],
Remark
1.10.4,
(ii)].
Although
at
first
glance,
it
might
appear
as
though
it
might
be
possible
to
develop
a
similar
theory
to
the
theory
developed
in
the
present
series
of
papers
based
on
a
more
general
sort
of
meromorphic
function
than
the
theta
function,
it
is
by
no
means
clear
that
such
a
more
general
meromorphic
function
satisfies
the
crucial
cyclotomic
rigidity,
discrete
rigidity,
and
constant
multiple
rigidity
properties
studied
in
[EtTh].
Of
these
properties,
the
cyclotomic
rigidity
property,
which
forms
the
basis
of
Corollary
1.10,
depends
most
explicitly
[cf.
[EtTh],
Remark
2.19.2]
on
ell
the
structure
of
the
theta
quotient
1
→
Δ
Θ
→
Δ
Θ
X
→
Δ
X
→
1
reviewed
in
[IUTchI],
Remark
3.1.2,
(iii)
[cf.
also
the
discussion
of
Remark
1.1.1
of
the
present
paper],
i.e.,
which
corresponds
to
the
“theta
group”
in
more
classical
treatments
of
the
theta
function.
Since
the
theta
function
is,
roughly
speaking,
essentially
characterized
among
meromorphic
functions
by
the
property
that
it
satisfies
the
“theta
symmetries”
arising
from
the
theta
group,
it
is
thus
difficult
to
see
how
to
generalize
the
theory
of
the
present
series
of
papers
so
as
to
treat
more
general
meromorphic
functions
than
the
theta
function
[cf.
Remark
1.1.1,
(v);
[IUTchIII],
Remark
2.3.3,
for
a
more
detailed
discussion
of
related
issues].
Also,
in
this
context,
it
is
useful
to
recall
that
unlike
the
theta
function
itself,
which
is
strictly
local
in
nature
[i.e.,
in
the
sense
that
it
is
only
defined,
a
priori,
at
v
∈
V
bad
],
the
theta
quotient
Δ
Θ
X
,
hence,
in
particular,
the
subquotient
Δ
Θ
,
is
defined
globally
[cf.
the
discussion
of
[IUTchI],
Remark
3.1.2]
over
the
various
number
fields
involved,
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
49
hence
may
be
applied
to
the
execution
of
various
global
anabelian
reconstruction
algorithms
via
the
“Θ-approach”
[cf.
[IUTchI],
Remark
3.1.2].
Corollary
1.11.
(Multiradial
MLF-Galois
Pair
Cyclotomic
Rigidity
Isomorphisms
with
Indeterminacies)
Write
(R,
C,
Φ
:
R
→
C)
—
i.e.,
in
the
notation
of
Example
1.8,
(viii),
(Π
M
TM
(Π),
G
O
gp
(G),
α
,×μ
)
→
(G
O
×μ
(G))
—
for
the
multiradial
environment
discussed
in
Example
1.8,
(viii).
Consider
(a)
the
Γ-orbit
[where
we
recall
that
Γ
⊆
Z
×
is
a
closed
subgroup]
∼
μ
Z
(G)
→
μ
Z
(O
×
(G))
=
Hom(Q/Z,
O
×
(G))
def
(∗
bs-Gal
G,
)
of
the
cyclotomic
rigidity
isomorphism
obtained
by
applying
to
the
MLF-Galois
pair
determined
by
G
O
(G)
the
algorithm
applied
to
∼
construct
[the
inverse
of
]
the
isomorphism
“μ
Z
(M
TM
)
→
μ
Z
(G)”
in
[Ab-
sTopIII],
Remark
3.2.1
[cf.
the
discussion
of
Proposition
1.3,
(ii)];
and
(b)
the
Aut(G)-orbit
[where
we
recall
from
[AbsAnab],
Proposition
1.2.1,
(vi),
that
Aut(G)
admits
a
natural
cyclotomic
character]
of
isomorphisms
∼
μ
Z
(G)
→
(l
·
Δ
Θ
)(Π)
(∗
bs-Gal
G,Π
)
obtained
by
composing
the
poly-isomorphism
induced
by
applying
“μ
Z
(−)”
to
the
[inverse
of
the]
full
poly-isomorphism
of
topological
groups
α
:
∼
Π/Δ
→
G
[cf.
Example
1.8,
(i)]
with
the
natural
isomorphism
“μ
Z
(G
k
)
∼
→
μ
Z
(Π
X
)”
of
[AbsTopIII],
Corollary
1.10,
(c)
[cf.
the
discussion
of
Proposition
1.3,
(ii)].
Then
the
data
consisting
of
the
triple
(Π,
G,
α)
[cf.
Example
1.8,
(i)],
the
topological
G-modules
constituted
by
the
domain
and
codomain
of
(∗
bs-Gal
G,
),
the
topological
Π-
bs-Gal
module
constituted
by
the
codomain
of
(∗
G,Π
),
and
the
poly-isomorphisms
(∗
bs-Gal
G,
)
bs-Gal
and
(∗
G,Π
)
determines
a
functor
R
→
F
which
arises
from
a
functorial
algorithm
in
the
triple
(Π,
G,
α);
denote
the
corresponding
graph
[cf.
Example
1.9,
(i)]
by
R
†
.
In
particular,
the
resulting
natural
functor
Ψ
R
:
R
→
R
†
[cf.
Example
1.9,
(i)]
is
multiradially
defined.
Proof.
The
various
assertions
of
Corollary
1.11
follow
immediately
from
the
defi-
nitions
involved.
Remark
1.11.1.
(i)
In
the
context
of
Corollary
1.11,
it
is
useful
to
recall
that:
(a)
the
group
of
automorphisms
of
the
underlying
ind-topological
monoid
equipped
with
a
topological
group
action
—
i.e.,
in
the
terminology
of
[AbsTopIII],
Definition
3.1,
(ii),
MLF-Galois
TM-pair
—
of
G
O
(G)
50
SHINICHI
MOCHIZUKI
maps
bijectively
[i.e.,
by
forgetting
“O
(G)”]
onto
the
group
of
automor-
phisms
of
the
topological
group
G
[cf.
[AbsTopIII],
Proposition
3.2,
(iv)];
(b)
the
group
of
automorphisms
of
the
underlying
ind-topological
module
equipped
with
a
topological
group
action
—
i.e.,
in
the
terminology
of
[AbsTopIII],
Definition
3.1,
(ii),
MLF-Galois
TCG-pair
—
of
G
O
×
(G)
maps
surjectively
[i.e.,
by
forgetting
“O
×
(G)”]
onto
the
group
of
auto-
morphisms
of
the
topological
group
G,
with
kernel
given
by
the
[G-linear]
automorphisms
of
[the
underlying
ind-topological
module
of]
O
×
(G)
de-
termined
by
the
natural
action
of
Z
×
[cf.
[AbsTopIII],
Proposition
3.3,
(ii)].
Also,
we
observe
that
by
the
same
proof
involving
the
Kummer
map
as
that
given
for
(b)
in
[AbsTopIII],
Proposition
3.3,
(ii),
it
follows
that
(c)
the
group
of
automorphisms
of
the
underlying
ind-topological
module
equipped
with
a
topological
group
action
of
G
O
gp
(G)
maps
surjectively
[i.e.,
by
forgetting
“O
gp
(G)”]
onto
the
group
of
auto-
morphisms
of
the
topological
group
G,
with
kernel
given
by
the
[G-linear]
automorphisms
of
[the
underlying
ind-topological
module
of]
O
gp
(G)
de-
termined
by
the
natural
action
of
Z
×
[or,
equivalently,
maps
bijectively
onto
the
group
of
automorphisms
of
the
underlying
ind-topological
mod-
ule
equipped
with
a
topological
group
action
of
G
O
×
(G)
—
cf.
(b)].
On
the
other
hand,
one
verifies
immediately
that
(d)
the
underlying
ind-topological
module
of
O
×μ
(G)
is
divisible,
hence
admits
a
natural
action
by
Q
p
.
In
particular,
if,
in
(b),
one
replaces
“O
×
”
by
“O
×μ
”,
then
the
resulting
description
of
the
kernel
is
false.
(ii)
In
the
present
series
of
papers,
we
shall
primarily
be
interested
in
Corollary
1.11
in
the
case
where
Γ
=
Z
×
.
That
is
to
say,
allowing
for
a
Γ
(=
Z
×
)-multiple
indeterminacy
corresponds
precisely
to
working,
in
the
case
of
G
O
×
(G),
with
the
underlying
ind-topological
module
equipped
with
topological
group
action
[cf.
(i),
(b)].
Remark
1.11.2.
(i)
Observe
that,
in
the
context
of
the
discussion
of
Remark
1.11.1,
(i),
(b),
the
natural
action
of
Z
×
on
[the
underlying
ind-topological
module
equipped
with
a
topological
group
action
of]
G
O
×
(G)
is
compatible
with
pull-back
via
the
∼
composite
of
the
natural
surjection
Π
Π/Δ
with
any
isomorphism
Π/Δ
→
G
[cf.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
51
the
notation
of
Example
1.8].
That
is
to
say,
one
has
a
natural
action
of
Z
×
on
[the
underlying
ind-topological
module
equipped
with
a
topological
group
action
of]
the
resulting
pair
Π
O
×
(G).
Observe,
moreover,
that
this
action
of
Z
×
fails
to
be
compatible
with
the
ring
structure
on
O
×
(G)⊗Q
[i.e.,
the
ring
structure
determined
by
applying
the
p-adic
logarithm].
That
is
to
say,
even
though
this
ring
structure
on
“O
×
”
may
[unlike
the
case
with
G!]
be
reconstructed
from
the
topological
group
Π
[cf.
[AbsTopIII],
Theorem
1.9],
the
natural
action
of
Z
×
on
Π
O
×
(G)
fails
to
preserve
the
ring
structure
reconstructed
from
Π.
(ii)
The
observations
of
(i)
are
of
interest
in
the
context
of
understanding
our
adoption
of
“G”
as
opposed
to
“Π”
in
the
construction
of
the
“Θ-link”
between
distinct
Θ-Hodge
theaters
given
in
[IUTchI],
Corollary
3.7.
That
is
to
say,
even
if
one
tries
to
“force
a
preservation
of
arithmetic
holomorphic
structures”
between
distinct
Θ-Hodge
theaters
by
working
with
“Π
O
×
(G)”
instead
of
“G
O
×
(G)”,
this
does
not
result
in
the
establishment
of
a
consistent
common
arithmetic
holomorphic
structure
for
distinct
Θ-Hodge
theaters,
since
the
establishment
of
such
a
consistent
common
arithmetic
holomorphic
structure
is
already
obstructed
by
the
fact
that
distinct
Θ-Hodge
theaters
only
share
a
common
“O
×
”
[cf.
[IUTchI],
Corollary
3.7,
(iii)]
—
on
which
Z
×
acts
[cf.
(i)]
—
i.e.,
as
opposed
to
a
common
“O
”.
Here,
we
recall
that
the
establishment
of
a
common
“O
”
is
obstructed,
in
a
quite
essential
manner,
by
the
“valuative
portion
†
Θ
v
→
‡
q
”
of
the
Θ-link
[cf.
[IUTchI],
Remark
v
3.8.1,
(i)].
Remark
1.11.3.
(i)
In
some
sense,
the
starting
point
of
any
discussion
of
radial
environments
is
the
description
of
the
radial
functor,
i.e.,
the
specification
of
“which
portion
of
the
radial
data
one
takes
for
one’s
coric
data”.
From
the
point
of
view
of
the
theory
of
[IUTchI],
§3
[cf.,
especially,
the
portion
at
v
∈
V
bad
of
[IUTchI],
Corollaries
3.7,
3.9],
the
coric
data
should,
in
particular,
include
the
quotient
Π
Π/Δ
∼
=
G
of
the
topological
group
Π
isomorphic
to
Π
tp
that
appears
in
a
Θ-Hodge
theater.
On
the
X
k
other
hand,
in
[IUTchIII],
we
shall
ultimately
be
interested
in
applying
the
theory
of
[AbsTopIII],
§3,
§5,
in
which
various
objects
[such
as
“Π
M
TM
(Π)”,
“G
O
(G)”,
“G
O
×
(G)”,
etc.]
are
constructed
group-theoretically
from
Π
or
G.
One
important
aspect
of
the
theory
of
[AbsTopIII],
§3,
§5,
is
that
after
these
objects
are
constructed
group-theoretically
from
Π
or
G,
one
then
proceeds
to
forget
the
“anabelian
structure”
of
these
objects,
i.e.,
one
forgets
the
data
consisting
of
the
way
in
which
these
objects
[such
as
MLF-Galois
TM-pairs,
MLF-Galois
TCG-
pairs,
etc.]
are
constructed
from
Π
or
G.
From
the
point
of
view
of
the
issue
of
“specification
of
coric
data”,
if
one
takes,
for
instance,
“G”
to
be
a
part
of
one’s
coric
data,
then
any
objects
constructed
group-theoretically
from
G
may
also
be
regarded
naturally
as
constituting
a
portion
of
the
coric
data
—
so
long
as
one
regards
these
objects
as
being
equipped
with
the
corresponding
“anabelian
structures”
[i.e.,
the
data
that
specifies
the
way
in
which
they
were
constructed
group-theoretically
from
G].
On
the
other
hand,
once
one
forgets
these
anabelian
structures,
it
is
no
longer
the
case
that
such
an
object
may
also
be
regarded
naturally
as
constituting
a
portion
of
the
coric
data.
That
is
to
say,
once
one
forgets
the
anabelian
structure
of
such
an
object,
it
is
necessary
to
specify
explicitly
that
such
an
object
is
to
52
SHINICHI
MOCHIZUKI
be
regarded
as
a
portion
of
the
coric
data,
i.e.,
as
a
portion
of
the
radial
data
that
is
to
be
subject
to
the
“gluing”,
or
“identification”,
discussed
in
Example
1.7,
(v).
(ii)
In
light
of
the
“coricity
of
O
×
”
given
in
[IUTchI],
Corollary
3.7,
(iii),
in
addition
to
“G”,
it
is
possible
to
take
the
underlying
MLF-Galois
TCG-pair
of
“G
O
×
(G)”
to
be
part
of
one’s
coric
data.
By
applying
Remark
1.11.1,
(i),
(b),
it
follows
that
this
amounts
to
working
with
“G
O
×
(G)”
up
to
an
(Aut(G),
Γ
(=
Z
×
))-indeterminacy
—
where
we
recall
from
Remark
1.8.1
that
the
p-adic
portion
of
the
Γ-indeterminacy
cannot
be
subsumed
into
the
Aut(G)-indeterminacy
[i.e.,
which
arises
from
the
fact
that
G
is
only
known
up
to
isomorphism
as
a
topological
group].
This
situation
is
precisely
the
situation
formulated
in
Example
1.8,
(iii).
On
the
other
hand,
as
we
saw
already
in
Corollary
1.10
[cf.
Remark
1.10.2],
and
as
we
shall
see
again
in
Corollary
1.12
below,
in
order
to
construct
certain
multiradially
defined
functors
that
will
be
of
substantial
importance
in
the
development
of
the
theory
of
the
present
series
of
papers,
it
is
necessary
to
form
the
quotient
of
“O
×
(−)”
by
its
torsion
subgroup
“O
μ
(−)”,
i.e.,
to
work
with
“O
×μ
(−)”,
rather
than
“O
×
(−)”.
Here,
we
note
[cf.
Example
1.8,
(ix);
Remark
1.11.1,
(i),
(d)]
that
one
does
not
wish
here
to
work
solely
with
the
underlying
ind-topological
module
equipped
with
topological
group
action
determined
by
“G
O
×μ
(G)”.
On
the
other
hand,
by
applying
[IUTchI],
Corollary
3.7,
(iii),
together
with
Remark
1.11.1,
(i),
(b),
one
concludes
that
it
is
possible
to
glue
together,
in
a
consistent
fashion,
the
various
“G
O
×μ
(G)”
[cf.
Fig.
1.4]
arising
from
distinct
Θ-Hodge
theaters
up
to
an
(Aut(G),
Γ
(=
Z
×
))-indeterminacy
[where
again
we
recall
from
Remark
1.8.1
that
the
p-adic
portion
of
the
Γ-indeter-
minacy
cannot
be
subsumed
into
the
Aut(G)-indeterminacy].
This
sort
of
situation
is
formulated
in
the
radial
environments
of
Example
1.8,
(v),
(vi),
(viii),
(ix)
[i.e.,
where
one
takes
“Γ
×μ
”
to
be
the
image
Im(Γ)
of
Γ].
One
important
point
in
this
context
is
that
even
if
one
takes
“G
O
×μ
(G)”
[i.e.,
as
opposed
to
“G
O
(G)”,
“G
O
gp
(G)”,
or
“G
O
×
(G)”]
as
one’s
coric
data,
the
condition
of
compatibility
with
respect
to
the
natural
maps
O
gp
(G)
←
O
×
(G)
O
×μ
(G)
[cf.
Example
1.8,
(viii)]
implies
that
the
(Aut(G),
Γ
(=
Z
×
))-indeterminacy
on
“G
O
×μ
(G)”
induces
a
(Aut(G),
Γ
(=
Z
×
))-indeterminacy
on
“G
O
×
(G)”
and
“G
O
gp
(G)”
—
where
one
may
think
of
the
“Γ-indeterminacy
on
O
gp
(G)”
as
representing
the
“Γ-indeterminacy
in
the
specification
of
the
submonoid
O
(G)
⊆
O
gp
(G)”.
It
is
precisely
these
indeterminacies
that
induce
the
indeterminacies
—
i.e.,
“orbits”
—
that
appear
in
Corollary
1.11,
(a),
(b),
in
sharp
contrast
to
the
“strict
cyclotomic
rigidity”
[i.e.,
without
any
indeterminacies]
of
Corollary
1.10
[cf.
Remark
1.10.2].
(iii)
Note
that
the
algorithms
applied
to
construct
cyclotomic
rigidity
iso-
morphisms
in
Corollaries
1.10
and
1.11,
(a),
are
obtained
by
composing
with
a
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
53
group-theoretic
construction
algorithm
an
algorithm
whose
input
data
is
“post-
anabelian”
—
i.e.,
consists
of
a
type
of
mathematical
object
that
arises
upon
forgetting
the
anabelian
structure
determined
by
the
group-theoretic
construction
algorithm.
More
concretely,
this
post-anabelian
input
data
consists
of
a
system
of
mono-theta
environments
in
the
case
of
Corollary
1.10
and
of
an
MLF-Galois
TM-
pair
in
the
case
of
Corollary
1.11,
(a).
As
discussed
in
(ii),
the
indeterminacies
that
act
on
this
post-anabelian
input
data
induce
the
indeterminacies
—
i.e.,
“orbits”
—
that
appear
in
Corollary
1.11,
(a),
(b).
Put
another
way,
(a)
the
indeterminacies
—
i.e.,
“orbits”
—
that
appear
in
Corollary
1.11,
(a),
(b),
are
a
consequence
of
the
highly
nontrivial
relationship
[cf.
the
dis-
cussion
of
(ii)]
between
the
input
data
“O
(−)”
of
the
cyclotomic
rigidity
algorithm
involved
and
the
coric
data
“O
×μ
(−)”.
By
contrast,
(b)
the
“strict
cyclotomic
rigidity”
asserted
in
Corollary
1.10
is
a
consequence
[cf.
Remark
1.10.2]
of
the
triviality
of
the
homomorphism
that
relates
the
cyclotomic
portion
of
“O
(−)”
—
which
is
the
only
portion
of
“O
(−)”
that
appears
in
a
mono-theta
environment
—
to
the
coric
data
“O
×μ
(−)”.
Here,
it
is
important
to
note
that
although
frequently
in
discussions
of
various
“re-
construction
algorithms”
[such
as
group-theoretic
reconstruction
algorithms],
em-
phasis
is
placed
on
the
existence
of
“some”
reconstruction
algorithm,
the
present
discussion
of
the
multiradiality
of
cyclotomic
rigidity
isomorphisms
in
the
con-
text
of
Corollaries
1.10,
1.11
yields
an
important
example
of
the
phenomenon
that
sometimes
not
only
the
existence
of
“some”
reconstruction
algorithm,
but
also
the
content
of
the
reconstruction
algorithm
[cf.
the
discussion
of
[IUTchIV],
Example
3.5]
is
of
crucial
importance
in
the
development
of
the
theory.
(iv)
Here,
we
note
in
passing
that
one
may
eliminate
the
(Aut(G),
Γ)-indeter-
minacy
of
Corollary
1.11,
(a),
(b),
by
working,
in
the
fashion
of
Example
1.9,
(iv),
(b),
with
uniradially
defined
functors
[that
is
to
say,
in
the
case
of
Corollary
1.11,
(a),
(b),
replacing
“G
O
(G)”
by
“Π
M
TM
(Π)”
and
“G”
by
“Π/Δ”
and
working
with
the
uniradial
environment
corresponding
to
the
assignment
×μ
(Π))
(Π
M
TM
(Π))
→
(Π/Δ
M
TM
—
i.e.,
for
which
the
definition
of
the
coric
data
coincides
with
the
definition
of
the
coric
data
of
the
multiradial
environment
in
the
statement
of
Corollary
1.11].
(v)
The
reason
[cf.
the
discussion
of
(iii)]
that
we
wish
to
consider
cyclotomic
rigidity
algorithms
whose
input
data
is
post-anabelian
is
that
we
wish
to
be
able
to
apply
the
same
algorithms
to
input
data
that
does
not
necessarily
arise
from
a
group-theoretic
construction
algorithm
—
e.g.,
to
input
data
that
arises
from
the
[divisor
and
rational
function]
monoid
portion
of
a
Frobenioid,
as
in
Proposition
1.3.
In
the
context
of
Proposition
1.3,
the
exterior
cyclotome
of
the
mono-theta
en-
vironment
that
appears
in
Corollary
1.10
and
the
cyclotome
arising
from
“O
(−)”
that
appears
in
Corollary
1.11,
(a),
both
correspond
to
the
same
cyclotome
“μ
N
(S)”
[which
arises
from
the
monoid
portion
of
the
Frobenioid
involved].
On
the
other
hand,
at
the
level
of
construction
algorithms,
in
order
to
relate
the
exterior
cyclo-
×
tome
“Π
μ
(M
Θ
∗
(Π))”
of
Corollary
1.10
to
the
cyclotome
“μ
Z
(O
(G))”
of
Corollary
54
SHINICHI
MOCHIZUKI
1.11,
(a),
it
is
necessary
[cf.
Proposition
1.3,
(iii)]
to
pass
through
the
cyclotome
“(l
·
Δ
Θ
)(Π)”
by
applying
the
cyclotomic
rigidity
isomorphisms
of
Corollaries
1.10,
1.11
—
which,
in
the
case
of
Corollary
1.11,
results
in
various
indeterminacies.
Put
another
way,
the
Frobenioid-theoretic
identification
[i.e.,
via
“μ
N
(S)”]
of
Proposi-
×
tion
1.3
between
the
cyclotomes
“Π
μ
(M
Θ
∗
(Π))”,
“μ
Z
(O
(G))”
of
Corollaries
1.10;
1.11,
(a),
may
be
thought
of
either
as
being
only
uniradially
defined
[cf.
(iv)],
or
as
multiradially
defined,
but
only
up
to
certain
indeterminacies.
Remark
1.11.4.
(i)
One
way
to
understand
the
significance
of
the
cyclotomic
rigidity
isomor-
phism
obtained
in
Corollary
1.10
—
i.e.,
of
the
triviality
of
the
homomorphism
that
relates
the
cyclotomic
portion
of
“O
(−)”
to
the
coric
data
“O
×μ
(−)”
[cf.
Remark
1.11.3,
(iii),
(b)]
—
relative
to
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.11,
which
involves
substantial
indeterminacies
arising
from
the
highly
nontrivial
re-
lationship
between
the
input
data
“O
(−)”
of
the
cyclotomic
rigidity
algorithm
involved
and
the
coric
data
“O
×μ
(−)”
[cf.
Remark
1.11.3,
(iii),
(a)],
is
as
a
sort
of
splitting,
or
decoupling,
that
serves
to
separate
the
“purely
radial
data”
that
appears
in
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.10
from
the
“purely
coric
data”
constituted
by
“O
×μ
(−)”.
This
point
of
view
is
discussed
further
in
Remark
1.12.2,
(vi),
below.
(ii)
From
the
point
of
view
of
the
discussion
of
Remark
1.9.2,
(iii),
the
“purely
radial
data”
that
appears
in
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.10
depends
on
the
tautological
collection
of
“labels
of
all
possible
arithmetic
holo-
morphic
structures”.
That
is
to
say,
the
algorithms
of
Corollary
1.10
do
not
give
rise
to
a
“detailed,
explicit
description”
of
these
labels
in
terms
of
the
“purely
coric
data
O
×μ
(−)”.
On
the
other
hand,
one
may
also
consider
a
modified
version
of
Corollary
1.10
in
which
(∗)
one
replaces
“O
×μ
(−)”
by
“O
×
(−)”
[i.e.,
so
that
the
crucial
triviality
discussed
in
Remark
1.11.3,
(iii),
(b),
no
longer
holds!]
and
applies
the
tautological
approach
discussed
in
Example
1.9,
(iv),
(a),
to
construct-
ing
the
cyclotomic
rigidity
isomorphism
[without
indeterminacies!]
under
consideration.
If
one
works
with
this
modified
version
(∗),
then
the
codomain
of
the
cyclotomic
rigidity
isomorphism
under
consideration
may
be
thought
of
as
the
submodule
“O
μ
(−)”
of
the
“purely
coric
data
O
×
(−)”,
equipped
with
a
“certain
rigidity”
that
depends
on
the
choice
of
an
element
of
the
collection
of
“labels
of
all
possible
arithmetic
holomorphic
structures”.
That
is
to
say,
whereas
Corollary
1.10
has
the
drawback
of
being
“not
entirely
free
of
label-dependence”,
the
significance
of
Corollary
1.10
[as
stated!]
relative
to
the
tautological
modified
version
(∗)
lies
in
the
fact
that
the
label-dependence
inherent
in
Corollary
1.10
is
confined
to
the
“purely
radial
component”
of
the
splitting,
or
decoupling,
discussed
in
(i)
—
i.e.,
unlike
the
case
with
(∗),
the
algorithms
of
Corollary
1.10
yield
a
“purely
coric
component”
that
is
free
of
such
“unwanted”
label-dependent
data.
Thus,
in
summary,
unlike
the
case
with
(∗),
the
algorithms
of
Corollary
1.10
yield
out-
put
data
equipped
with
a
splitting,
or
decoupling,
into
label-dependent
[i.e.,
“purely
radial”]
and
label-independent
[i.e.
“purely
coric”]
components.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
55
Remark
1.11.5.
Suppose
that
we
are
in
the
situation
of
Corollary
1.11.
(i)
Recall
the
natural
surjection
H
1
(G,
μ
Z
(G))
Z
—
which
is
constructed
via
a
functorial
group-theoretic
algorithm
in
[AbsTopIII],
Corollary
1.10,
(b).
That
is
to
say,
when
G
=
G
k
,
this
surjection
is
the
surjection
determined
by
the
valuation
of
k
on
the
image
of
the
natural
Kummer
map
k
×
→
H
1
(G
k
,
μ
Z
(G
k
))
—
where
we
recall
that
the
image
of
this
Kummer
map
is
equal
to
the
inverse
image
of
Z
⊆
Z
via
the
surjection
under
consideration.
In
particular,
the
existence
of
this
functorial
group-theoretic
algorithm
implies
that
the
data
consisting
of
this
natural
surjection
—
hence,
in
particular,
its
kernel,
i.e.,
“O
k
×
”
—
may
be
formulated
as
a
corically,
hence,
in
particular,
as
a
multiradially
[cf.
Example
1.7,
(iv)],
defined
functor.
[We
leave
the
routine
details
to
the
reader.]
(ii)
On
the
other
hand,
if
one
applies
the
isomorphisms
(∗
bs-Gal
G,
)
[cf.
also
the
bs-Gal
poly-isomorphism
α
of
Example
1.8,
(ii)]
and
(∗
G,Π
),
of
Corollary
1.11,
then
the
natural
surjection
of
(i)
gives
rise
to
natural
surjections
H
1
(G,
μ
Z
(M
TM
(Π)))
Z;
H
1
(G,
(l
·
Δ
Θ
)(Π))
Z
—
which
yield
data
that
may
be
formulated
either
as
a
uniradially
defined
functor
[cf.
Remark
1.11.3,
(iv)]
or,
when
considered
up
to
a
Z
×
-indeterminacy,
as
a
multiradially
defined
functor
[cf.
Corollary
1.11].
In
particular,
the
kernels
of
these
natural
surjections
yield
data
that
may
be
formulated
as
a
multiradially
defined
functor.
[We
leave
the
routine
details
to
the
reader.]
Remark
1.11.6.
The
importance
of
cyclotomic
rigidity
in
the
theory
of
the
present
series
of
papers
is
interesting
in
light
of
the
analogy
between
the
ideas
of
the
present
series
of
papers
and
the
p-adic
Teichmüller
theory
of
[pTeich]
[cf.
the
discussion
of
[AbsTopIII],
§I5].
Indeed,
the
proof
of
a
fundamental
absolute
p-adic
anabelian
result
concerning
the
canonical
curves
that
arise
in
the
theory
of
[pTeich]
[cf.
[CanLift],
Theorem
3.6]
depends,
in
an
essential
way,
on
a
certain
cyclotomic
rigidity
result
proven
in
an
earlier
paper
[cf.
[AbsAnab],
Lemma
2.5,
(ii)].
In
this
context,
we
observe
that
one
important
theme
that
appears
both
in
the
present
series
of
papers
and
in
the
theory
of
[CanLift],
§3,
is
the
idea
that
cyclotomes
should
be
thought
of
as
the
“skeleta
of
arithmetic
holomorphic
structures”
—
cf.
the
relation
of
S
1
to
C
×
in
the
complex
archimedean
theory.
We
are
now
ready
to
discuss
the
main
result
of
the
present
§1.
Corollary
1.12.
(Multiradial
Constant
Multiple
Rigidity)
Write
(R,
C,
Φ
:
R
→
C)
—
i.e.,
in
the
notation
of
Example
1.8,
(v),
(vi),
×μ
(G),
α
μ,×μ
)
→
(G
O
×μ
(G))
(Π
Π
μ
(M
Θ
∗
(Π))
⊗
Q/Z,
G
O
56
SHINICHI
MOCHIZUKI
—
for
the
multiradial
environment
discussed
in
Example
1.8,
(v),
(vi),
where
def
we
take
Γ
×μ
=
Ism.
Consider
the
functorial
algorithm
that
associates
to
Π
the
following
commutative
diagram
(†
×θ
)(Π)
×
×
1
M
TM
(Π)
M
TM
·
∞
θ(Π)
→
lim
−→
J
H
(Π
Ÿ
(Π)|
J
,
(l
·
Δ
Θ
)(Π))
⏐
⏐
⏐
⏐
×
×
1
Θ
Θ
M
TM
(M
Θ
M
TM
·
∞
θ
env
(M
Θ
∗
(Π))
∗
(Π))
→
lim
−→
J
H
(Π
Ÿ
(M
∗
(Π))|
J
,
Π
μ
(M
∗
(Π)))
—
where
(a)
J
ranges
over
the
finite
index
open
subgroups
of
Π;
“|
J
”
denotes
the
fiber
product
“×
Π
J”;
(b)
the
right-hand
vertical
arrow
is
the
isomorphism
of
modules
induced
by
the
cyclotomic
rigidity
isomorphism
obtained
via
the
functorial
algorithm
of
Corollary
1.10;
(c)
we
recall
that
it
follows
from
the
definitions
[cf.
Example
1.8,
(ii),
(iii);
[AbsTopIII],
Definition
3.1,
(vi);
[IUTchI],
Remark
3.1.2]
that
one
has
a
×
1
natural
inclusion
M
TM
(Π)
→
lim
J
H
(J,
(l
·
Δ
Θ
)(Π)),
hence
a
natural
−
→
×
(Π)
into
the
inductive
limit
of
the
first
line;
inclusion
of
M
TM
×
(M
Θ
(d)
we
define
M
TM
∗
(Π))
and
the
left-hand
vertical
arrow
to
be
the
sub-
module
and
bijection
induced
by
the
cyclotomic
rigidity
isomorphism
of
(b);
×
×
·
∞
θ(Π)
=
M
TM
(Π)
·
∞
θ(Π);
here,
∞
θ(Π)
is
obtained
via
(e)
we
define
M
TM
the
functorial
algorithm
of
Proposition
1.4,
applied
to
Π,
and
the
“·”
is
to
be
understood
as
being
taken
with
respect
to
the
module
structure
[i.e.,
which
is
usually
denoted
additively!]
of
the
ambient
cohomology
module;
def
×
×
Θ
Θ
·
∞
θ
env
(M
Θ
(f)
we
define
M
TM
∗
(Π))
=
M
TM
(M
∗
(Π))
·
∞
θ
env
(M
∗
(Π));
here,
Θ
∞
θ
env
(M
∗
(Π))
is
obtained
via
the
functorial
algorithm
of
Proposition
1.5,
(iii),
applied
to
M
Θ
∗
(Π)
[cf.
Propositions
1.2,
(i);
1.5,
(i)];
the
“·”
is
as
in
(e);
def
(g)
the
horizontal
arrows
“→”
are
the
natural
inclusions.
×μ
μ
μ
×
×
(−)
=
M
TM
(−)/M
TM
(−),
where
M
TM
(−)
⊆
M
TM
(−)
de-
Also,
let
us
write
M
TM
notes
the
submodule
of
torsion
elements.
Then:
def
(i)
There
is
a
functorial
group-theoretic
algorithm
Π
→
{(ι,
D)}(Π)
def
that
assigns
to
the
topological
group
Π
a
collection
of
pairs
(ι,
D)
—
where
Δ
Ÿ
(Π)
=
Π
Ÿ
(Π)
Δ,
ι
is
a
Δ
Ÿ
(Π)-outer
automorphism
of
Π
Ÿ
(Π)
[cf.
Proposition
1.4],
and
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
57
[by
abuse
of
notation]
D
⊆
Π
Ÿ
(Π)
is
a
Δ
Ÿ
(Π)-conjugacy
class
of
closed
subgroups
—
with
the
property
that
when
Π
=
Π
tp
X
,
the
resulting
collection
of
pairs
coincides
k
with
the
collection
of
“pointed
inversion
automorphisms”
of
Remark
1.4.1,
(ii).
Here,
each
pair
(ι,
D)
will
be
referred
to
as
a
pointed
inversion
automorphism.
If
(ι,
D)
is
a
pointed
inversion
automorphism,
and
ι
induces
an
“action
up
to
torsion”
on
some
subset
“(−)”
of
an
abelian
group
[i.e.,
an
action
on
the
image
of
this
subset
in
the
quotient
of
the
abelian
group
by
its
torsion
subgroup],
then
we
shall
denote
by
a
superscript
“ι”
on
“(−)”
the
subset
of
ι-invariants
with
respect
to
this
“action
up
to
torsion”,
i.e.,
the
subset
of
“(−)”
that
consists
precisely
of
those
elements
of
“(−)”
whose
images
in
the
quotient
of
the
abelian
group
by
its
torsion
subgroup
are
fixed
by
the
induced
action
of
ι.
(ii)
Let
(ι,
D)
be
a
pointed
inversion
automorphism
associated
to
Π
[cf.
(i)].
Then
restriction
to
the
subgroup
D
⊆
Π
Ÿ
(Π)
determines
[the
horizontal
arrows
in]
a
commutative
diagram
×
·
∞
θ(Π)}
ι
{M
TM
⏐
⏐
−→
×
M
TM
(Π)
⏐
⏐
×
ι
{M
TM
·
∞
θ
env
(M
Θ
∗
(Π))}
−→
×
M
TM
(M
Θ
∗
(Π))
1
⊆
lim
−→
J
H
(J,
(l
·
Δ
Θ
)(Π))
1
Θ
⊆
lim
−→
J
H
(J,
Π
μ
(M
∗
(Π)))
—
where
J
ranges
over
the
finite
index
open
subgroups
of
Π
[cf.
(a)];
the
vertical
arrows
are
the
isomorphisms
induced
by
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.10
[cf.
(b)].
Here,
the
inverse
images
of
the
submodules
of
torsion
μ
(−)”
elements
—
i.e.,
[up
to
various
natural
isomorphisms]
the
submodules
“M
TM
—
via
the
upper
and
lower
horizontal
arrows
are
given,
respectively,
by
∞
θ(Π)
ι
and
Θ
ι
∞
θ
env
(M
∗
(Π))
.
In
particular,
we
obtain
a
functorial
algorithm
[in
the
topological
group
Π]
for
constructing
splittings
×μ
μ
(Π)
×
{
∞
θ(Π)
ι
/M
TM
(Π)};
M
TM
×μ
μ
Θ
ι
Θ
(M
Θ
M
TM
∗
(Π))
×
{
∞
θ
env
(M
∗
(Π))
/M
TM
(M
∗
(Π))}
(†
μθ
)(Π)
—
i.e.,
direct
product
decompositions
inside
the
quotients
of
the
inductive
limits
μ
(−)”
—
of
the
respective
on
the
right-hand
side
of
the
diagram
(†
×θ
)(Π)
by
“M
TM
×
×
ι
Θ
ι
images
of
{M
TM
·
∞
θ(Π)}
,
{M
TM
·
∞
θ
env
(M
∗
(Π))}
in
these
quotients.
(iii)
Consider
the
assignment
that
associates
to
the
data
×μ
(Π
Π
μ
(M
Θ
(G),
α
μ,×μ
)
∗
(Π))
⊗
Q/Z,
G
O
the
data
consisting
of
·
M
Θ
∗
(Π)
—
i.e.,
the
projective
systems
of
mono-theta
environ-
ments
of
Propositions
1.2,
(i);
1.5,
(i);
·
(†
×θ
)(Π)
—
i.e.,
“subsets”;
·
(†
μθ
)(Π)
—
i.e.,
“splittings”;
58
SHINICHI
MOCHIZUKI
·
the
diagram
Π
μ
(M
Θ
∗
(Π))
⊗
Q/Z
→
∼
μ
M
TM
(M
Θ
∗
(Π))
→
→
×μ
(Π)
M
TM
O
×μ
(G)
∼
→
∼
μ
M
TM
(Π)
(†
μ,×μ
)
∼
—
where
the
first
“
→
”
is
the
isomorphism
determined
by
the
injection
of
∼
Remark
1.5.2;
the
second
“
→
”
is
the
isomorphism
determined
by
the
ver-
tical
arrows
of
(†
×θ
)(Π);
the
“→”
is
the
trivial
homomorphism;
the
final
∼
“
→
”
denotes
the
poly-isomorphism
induced
by
the
poly-isomorphism
“α
×
”
of
Example
1.8,
(iii)
[cf.
also
the
discussion
of
“Γ
×μ
”
in
Example
1.8,
(iv)].
Then
this
assignment
determines
a
functor
R
→
F
which
arises
from
a
functo-
rial
algorithm;
denote
the
corresponding
graph
[cf.
Example
1.9,
(i)]
by
R
†
.
In
particular,
the
resulting
natural
functor
Ψ
R
:
R
→
R
†
[cf.
Example
1.9,
(i)]
is
multiradially
defined.
Proof.
Assertion
(i)
follows
immediately
from
the
discussion
of
Remark
1.4.1
and
the
references
quoted
in
this
discussion.
Assertion
(ii)
follows
immediately
from
the
structure
of
the
objects
under
consideration,
as
described
in
[EtTh],
Proposition
1.5,
(ii),
(iii)
[cf.
also
the
proofs
of
[EtTh],
Theorems
1.6,
1.10].
Finally,
the
multiradiality
of
assertion
(iii)
follows
immediately
from
the
characteristic
nature
μ
(−)”
that
appear
[cf.
the
discussion
of
of
the
various
torsion
submodules
“M
TM
Remark
1.10.2;
the
discussion
of
Remark
1.12.2
below].
Remark
1.12.1.
One
verifies
immediately
that
Corollaries
1.10,
1.11,
and
1.12
admit
“log-shell
versions”
[cf.
Example
1.8,
(ix)].
The
various
interpretations
of
these
corollaries
discussed
in
the
remarks
following
the
corollaries
also
apply
to
such
“log-shell
versions”.
Remark
1.12.2.
(i)
Modulo
the
multiradiality
of
the
cyclotomic
rigidity
isomorphism
of
Corol-
lary
1.10
[cf.
Corollary
1.12,
(b)],
the
essential
content
of
the
multiradiality
of
Corollary
1.12
lies
in
the
functorial
group-theoretic
algorithm
implicit
in
the
proof
of
[EtTh],
Theorem
1.10,
(i),
for
constructing
θ(Π)
up
to
a
μ
2l
-indeterminacy
—
i.e.,
as
opposed
to
only
up
to
a
“O
k
×
-indeterminacy”,
as
is
done
in
the
proof
of
[EtTh],
Theorem
1.6,
(iii)
—
together
with
the
[elementary]
observation
that
the
submodule
of
[any
isomorph
of]
O
k
×
constituted
by
the
2l-torsion
is
characteristic
[cf.
the
proof
of
Corollary
1.12,
(iii)].
That
is
to
say,
it
is
this
“essential
content”
that
implies
that
the
crucial
splittings
(†
μθ
)(Π)
are
compatible
with
gluing
together
the
various
collections
of
coric
data
“(G
O
×μ
(G))”
that
arise
from
distinct
arithmetic
holomorphic
structures.
(ii)
Here,
we
recall
in
passing
[cf.
also
the
discussion
of
Remark
1.4.1]
that
the
functorial
group-theoretic
algorithm
implicit
in
the
proof
of
[EtTh],
Theorem
1.10,
(i),
for
constructing
θ(Π)
up
to
a
μ
2l
-indeterminacy
consists
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
59
normalizing
the
étale
theta
functions
under
consideration
by
requiring
that
their
values
at
points
[cf.
also
the
discussion
of
Remark
1.12.4
below]
lying
over
the
2-torsion
point
“μ
−
”
of
[IUTchI],
Example
4.4,
(i),
be
∈
μ
2l
—
i.e.,
of
considering
étale
theta
functions
“of
standard
type”
[cf.
[EtTh],
Definition
1.9,
(ii);
[EtTh],
Theorem
1.10,
(i);
[EtTh],
Definition
2.7].
Also,
we
recall
from
the
proof
of
[EtTh],
Theorem
1.10,
(i),
that
the
decomposition
groups
⊆
Π
corre-
sponding
to
these
points
lying
over
the
2-torsion
point
“μ
−
”
are
reconstructed
by
applying,
among
other
tools,
the
elliptic
cuspidalizations
reviewed
in
Proposition
1.6,
(ii)
[cf.
also
the
discussion
of
Corollary
2.4,
(ii),
(b),
below].
(iii)
By
contrast,
if,
in
the
context
of
the
discussion
of
(i),
the
normalization
reviewed
in
(ii)
consisted
of
the
requirement
that
certain
values
of
the
étale
theta
function
be
equal,
for
instance,
to
2
=
1
+
1
∈
O
k
×
⊆
(k
×
)
∧
def
[where
we
recall
that
the
residue
characteristic
of
k
is
assumed
to
be
odd
—
cf.
[IUTchI],
Definition
3.1,
(b)],
i.e.,
an
element
of
(k
×
)
∧
whose
construction
depends,
in
an
essential
way,
on
the
ring
structure
relative
to
some
specific
Θ
±ell
NF-Hodge
theater
—
i.e.,
some
specific
arithmetic
holomorphic
structure
—
then
the
normal-
ization
would
fail
to
give
rise
to
a
multiradially
defined
functor,
although
[cf.
[AbsTopIII],
Corollary
1.10,
(h);
[IUTchI],
Remark
3.1.2]
it
would
nonetheless
give
rise
to
a
uniradially
defined
functor
[cf.
the
discussion
of
Example
1.9,
(iv),
(b);
Remark
1.11.5,
(ii)].
(iv)
From
the
point
of
view
of
the
further
development
of
the
theory
of
the
present
series
of
papers,
the
significance
of
obtaining
“splittings
up
to
a
μ-indeter-
minacy”
may
be
summarized
as
follows.
Ultimately,
we
shall
be
interested,
in
[IUTchIII],
in
applying
the
theory
of
log-shells
developed
in
[AbsTopIII]
[cf.
Remark
1.12.1].
From
the
point
of
view
of
log-shells,
which
may
be
thought
of
as
being
contained
in
O
×μ
(G),
an
indeterminacy
up
to
some
larger
subgroup
of
O
k
×
—
such
as,
for
instance,
the
subgroup
generated
by
2
=
1
+
1,
together
with
its
Aut(G)-
conjugates
[cf.
the
discussion
of
(iii)]
—
would
imply
that
one
may
only
work,
in
an
inconsistent
fashion,
with
[for
instance,
the
image
of
the
log-shell
in]
the
quotient
of
O
×μ
(G)
by
such
a
larger
subgroup
—
a
situation
which
is
unacceptable
from
the
point
of
view
of
the
further
develop-
ment
of
the
theory
of
the
present
series
of
papers.
(v)
The
discussion
in
(i),
(ii),
and
(iii)
above
of
the
multiradiality
of
the
crucial
splittings
(†
μθ
)(Π)
of
Corollary
1.12,
(ii),
yields
another
important
example
[cf.
Remark
1.11.3,
(iii)]
of
the
phenomenon
that
sometimes
not
only
the
existence
of
a
single
reconstruction
algorithm,
but
also
the
content
of
the
reconstruction
algorithm
is
of
crucial
importance
in
the
development
of
the
theory.
Similar
ideas,
relative
to
the
point
of
view
of
the
theory
of
[EtTh],
may
also
be
seen
in
the
discussion
of
[EtTh],
Remarks
1.10.2,
1.10.4.
(vi)
In
general,
multiradiality
amounts
to
a
sort
of
“surjectivity”
[cf.
the
defi-
nition
of
a
multiradial
environment
via
a
“fullness”
condition
in
Example
1.7,
(ii);
60
SHINICHI
MOCHIZUKI
the
discussion
of
Example
1.7,
(v)]
of
the
radial
data
onto
the
coric
data.
From
this
point
of
view,
the
content
of
the
multiradiality
of
the
splittings
(†
μθ
)(Π)
of
Corol-
lary
1.12,
(ii),
may
be
thought
of
as
consisting
of
a
splitting
of
this
“surjection
of
radial
data
onto
coric
data”
into
μ
(Π)},
(a)
a
“purely
radial
component”
constituted
by
{
∞
θ(Π)
ι
/M
TM
μ
Θ
ι
Θ
{
∞
θ
env
(M
∗
(Π))
/M
TM
(M
∗
(Π))}
and
×μ
×μ
(Π),
M
TM
(M
Θ
(b)
a
“purely
coric
component”
constituted
by
M
TM
∗
(Π))
[cf.
the
discussion
of
Remark
1.11.4].
Remark
1.12.3.
From
the
point
of
view
of
the
discussion
of
Remark
1.11.3,
it
×
×
is
useful
to
note
that
the
subsets
M
TM
·
∞
θ(Π),
M
TM
·
∞
θ
env
(M
Θ
∗
(Π))
that
appear
in
Corollary
1.12
may
be
thought
of
as
[“roots”
of]
the
images,
via
the
Kummer
map,
of
a
certain
generating
subset
of
the
monoid
of
rational
functions
“O
C
Θ
(−)”
v
defined
in
[IUTchI],
Example
3.2,
(v),
which
is
used
to
construct
the
underlying
Frobenioid
of
the
split
Frobenioid
“F
v
Θ
”
—
cf.
also
the
discussion
of
Kummer
classes
in
[EtTh],
Proposition
5.2,
(iii).
Here,
the
splittings
(†
μθ
)(Π)
of
Corollary
1.12,
(ii),
correspond
to
the
splitting
data
of
this
split
Frobenioid
F
v
Θ
.
Put
another
way,
this
monoidal
data
that
gives
rise
to
the
split
Frobenioid
F
v
Θ
may
be
thought
of
as
the
result
of
forgetting
the
“anabelian
struc-
×
×
·
∞
θ(Π),
M
TM
·
∞
θ
env
(M
Θ
ture”
of
M
TM
∗
(Π)),
and
(†
μθ
)(Π)
—
cf.
the
discussion
of
Remark
1.11.3,
(i),
(ii);
the
theory
of
§3
below,
especially,
Proposition
3.4.
In
particular,
the
specification
of
coric
data
“(G
O
×μ
(G))”
in
the
multiradial
environment
that
appears
in
Corollary
1.12
arises
naturally
from
the
point
of
view
of
applying
the
“coricity
of
O
×
”
given
in
[IUTchI],
Corollary
3.7,
(iii),
as
in
the
discussion
of
Remark
1.11.3,
(ii).
Finally,
we
recall
from
the
discussion
of
Remark
1.11.3,
(ii),
that
this
specification
of
coric
data
“(G
O
×μ
(G))”
has
the
effect
of
inducing,
in
particular,
an
(Aut(G),
Im(
Z
×
)
(⊆
Ism))-indeterminacy
on
“G
O
×μ
(G)”
[cf.
Corollary
1.12,
(iii)].
Remark
1.12.4.
The
fact
that
the
“theta
evaluation”
functorial
algorithm
of
Corollary
1.12,
(ii),
given
by
restriction
to
the
decomposition
groups
associated
to
the
point
“μ
−
”
involves
only
the
topological
group
“Π”
as
input
data
will
be
of
crucial
importance
when
we
combine
the
theory
developed
in
the
present
paper
with
the
theory
of
log-shells
[cf.
[AbsTopIII]]
in
[IUTchIII].
At
this
point,
it
is
useful
to
stop
and
consider
to
what
extent
this
sort
of
“group-theoretic
evaluation
algorithm”
is
an
inevitable
consequence
of
various
natural
conditions.
To
this
end,
let
us
suppose
that
we
are
given
some
“mysterious
evaluation
algorithm”
(abstract
theta
function)
→
(theta
values)
—
i.e.,
which
is
not
necessarily
given
by
restriction
to
the
decomposition
group
associated
to
a
closed
point.
Then
[cf.
[EtTh],
Remark
1.10.4;
the
theory
of
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
61
“log-wall”,
as
discussed
in
[AbsTopIII],
§I4]
it
is
natural
to
require
[cf.,
especially,
the
point
of
view
of
the
discussion
of
Remark
1.12.3]
that
this
algorithm
be
compatible
with
the
operation
of
forming
Kummer
classes
by
extract-
ing
N
-th
roots
of
the
“abstract
theta
function”
and
the
“theta
values”.
In
particular,
it
is
natural
to
require
that
this
algorithm
extend
to
coverings
[e.g.,
Galois
coverings]
on
both
the
input
and
output
data
of
the
algorithm.
But
then
the
natural
requirement
of
functoriality
with
respect
to
the
Galois
groups
on
either
side
leads
one
[cf.
Fig.
1.5
below],
in
effect,
to
the
conclusion
—
which
we
shall
refer
to
as
the
principle
of
Galois
evaluation
—
that
the
“mysterious
evaluation
algorithm”
under
consideration
in
fact
arises
from
a
section
G
→
Π
Ÿ
(Π)
of
the
natural
surjection
Π
Ÿ
(Π)
G.
Moreover,
by
the
“Section
Conjecture”
of
anabelian
geometry,
one
expects
that
such
[continuous]
sections
G
→
Π
Ÿ
(Π)
necessarily
arise
from
geometric
points.
[Here,
we
pause
to
observe
that
this
relation
to
the
“Section
Conjecture”
is
interesting
in
light
of
the
discussion
of
[IUTchI],
Example
4.5,
(i);
[IUTchI],
Remark
2.5.1.]
In
this
context,
it
is
useful
to
recall
that
from
the
point
of
view
of
the
theory
of
[AbsTopIII]
[cf.,
e.g.,
the
discussion
of
[AbsTopIII],
§I5],
the
group-theoreticity
of
the
evaluation
algorithm
may
be
thought
of
as
a
sort
of
abstract
analogue
of
the
condition,
in
the
p-adic
theory,
that
an
operation
involving
various
Frobenius
crystals
be
compatible
with
the
Frobenius
crystal
structures
[i.e.,
connection
and
Frobenius
action]
on
the
input
and
output
data
of
the
operation.
Π
Ÿ
(Π)
geometric
object
(+
coverings!)
that
support(s)
the
abstract
theta
function
geometric
object
(+
coverings!)
that
support(s)
the
theta
values
---
>
G
Fig.
1.5:
Theta
evaluation
and
Galois
functoriality
Remark
1.12.5.
(i)
Recall
that
the
scheme-theoretic
Hodge-Arakelov
theory
reviewed
in
[HA-
SurI],
[HASurII]
may
be
thought
of
as
a
sort
of
scheme-theoretic
version
of
the
well-known
classical
computation
of
the
Gaussian
integral
∞
e
−x
dx
=
2
√
π
−∞
—
i.e.,
by
thinking
of
the
square
of
this
integral
as
an
integral
over
the
cartesian
plane
R
2
,
which
may
be
computed
easily
by
applying
a
coordinate
transformation
into
polar
coordinates.
That
is
to
say
[cf.
the
left-hand
and
middle
columns
of
Fig.
1.6
below],
the
main
theorem
of
scheme-theoretic
Hodge-Arakelov
theory
is
a
certain
comparison
isomorphism
[cf.
[HASurI],
Theorem
A]
between
a
“de
Rham
side”
—
62
SHINICHI
MOCHIZUKI
which
consists
of
certain
sections
of
an
ample
line
bundle
on
the
universal
extension
of
an
elliptic
curve
—
and
an
“étale
side”
—
which
consists
of
arbitrary
functions
on
the
set
of
l-torsion
points
of
the
elliptic
curve
[where
l
is,
say,
some
odd
prime
number].
Here,
the
module
on
the
de
Rham
side
is
equipped
with
a
natural
Hodge
filtration,
while
the
module
on
the
étale
side
is
equipped
with
a
natural
Galois
action
by
GL
2
(F
l
).
The
ordered,
“step-like”
structure
of
the
Hodge
filtration
is
reminiscent
of
the
cartesian
structure
of
the
plane
R
2
,
i.e.,
regarded
as
an
ordered
collection
[parametrized
by
one
factor
of
R
2
]
of
lines
[corresponding
to
the
other
factor
of
R
2
].
On
the
other
hand,
the
GL
2
(F
l
)-symmetry
of
the
étale
side
is
reminiscent
of
the
rotational
symmetry
of
the
representation
of
the
Gaussian
integral
on
the
plane
via
2
Gaussian
polar
coordinates.
Moreover,
the
function
“e
−x
”
itself
appears
in
the
√
poles
that
appear
in
the
de
Rham
side
[cf.
[HASurI],
§1.1],
while
the
“
π”
may
be
thought
of
as
corresponding
to
the
[negative]
tensor
powers
of
the
sheaf
“ω”
of
invariant
differentials
on
the
elliptic
curve
that
appear
in
the
subquotients
of
the
Hodge
filtration,
which
give
rise
to
a
Kodaira-Spencer
isomorphism
[cf.
[HASurII],
Theorems
2.8,
2.10]
between
ω
⊗2
and
the
restriction
to
the
base
scheme
of
the
sheaf
of
logarithmic
differentials
on
the
moduli
stack
of
elliptic
curves
—
i.e.,
between
ω
and
the
“square
root”
of
this
sheaf
of
logarithmic
differentials.
Finally,
we
recall
that
this
relationship
between
the
theory
of
[HASurI],
[HASurII]
and
the
classical
Gaussian
integral
may
be
seen
more
explicitly
when
this
theory
is
restricted
to
the
archimedean
primes
of
a
number
field
via
the
“Hermite
model”
[cf.
[HASurI],
§1.1].
classical
Gaussian
integral
scheme-theoretic
Hodge-Arakelov
theory
inter-universal
Teichmüller
theory
cartesian
coordinates
de
Rham
side,
Hodge
filtration
Frobenius-like
structures,
Frobenius-picture
polar
coordinates
étale
side,
Galois
action
on
torsion
points
étale-like
structures,
étale-picture
Fig.
1.6:
Analogy
with
the
classical
Gaussian
integral
(ii)
Just
as
the
theory
of
[HASurI],
[HASurII]
may
be
thought
of
as
a
scheme-
theoretic
version
of
the
classical
theory
of
the
Gaussian
integral,
the
“inter-universal
Teichmüller
theory”
developed
in
the
present
se-
ries
of
papers
may
be
thought
of
as
a
sort
of
global
arithmetic/Galois-
theoretic
version
of
the
classical
Gaussian
integral
—
cf.
the
right-hand
column
of
Fig.
1.6.
Indeed,
the
ordered,
“step-like”
nature
of
the
cartesian
representation
of
the
Gaussian
integral
on
the
plane
is
remi-
niscent
of
the
structure
of
the
Frobenius-picture
discussed
in
[IUTchI],
Corollary
3.8;
[IUTchI],
Remark
3.8.1
—
i.e.,
in
particular,
of
the
notion
of
a
Frobenius-
like
mathematical
structure
that
appears
in
the
discussion
of
[FrdI],
Introduction.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
63
On
the
other
hand,
the
rotational
symmetry
of
the
representation
of
the
Gaussian
integral
on
the
plane
via
polar
coordinates
is
reminiscent
of
the
étale-picture
dis-
cussed
in
[IUTchI],
Corollary
3.9,
and
the
following
remarks
—
i.e.,
in
particular,
of
the
notion
of
an
étale-like
mathematical
structure
that
appears
in
the
discus-
sion
of
[FrdI],
Introduction.
The
étale-picture
that
arises
from
the
multiradially
defined
functor
of
Corollary
1.12
is
depicted
in
Fig.
1.7
below
[where
we
recall
the
notation
of
Proposition
1.4;
Example
1.8,
(iv)].
From
the
point
of
view
of
the
clas-
sical
series
representation
of
a
theta
function
—
i.e.,
roughly
speaking,
the
series
2
“
n∈Z
q
n
·
U
n
”
[cf.
[EtTh],
Proposition
1.4]
—
this
étale-picture
of
various
copies
of
the
Gaussian
function
“q
n
”
de-
fined
on
spokes
emanating
from
a
single
common
core
2
i
∞
θ(
Π)
⏐
⏐
...
∞
θ(
i
Π)
−→
...
mono-analytic
core
←−
∞
θ(
i
Π)
G
O
×μ
(G)
Ism
⏐
⏐
...
∞
θ(
i
...
Π)
Fig.
1.7:
Multiradial
étale
theta
functions
[cf.
also
the
point
of
view
of
Remark
1.12.2,
(vi)]
is
highly
reminiscent
of
the
polar
coordinate
representation
of
the
Gaussian
integral
on
the
plane.
In
this
context,
it
is
also
of
interest
to
observe
that
the
coordinate
transformation
e
−r
2
u
that
appears
in
the
radial
portion
of
the
integrand
of
the
Gaussian
integral
that
arises
from
the
transformation
from
cartesian
to
polar
coordinates
−x
2
−y
2
−r
2
2
e
dx
dy
=
e
·
2rdr
dθ
2
·
(
e
−x
dx)
2
=
2
·
2
=
d(e
−r
)
dθ
=
du
dθ
is
formally
reminiscent
of
the
Θ-link
“
†
Θ
v
→
‡
q
”
[cf.
[IUTchI],
Remark
3.8.1,
v
(i)],
various
versions
of
which
play
a
central
role
in
the
theory
of
the
present
series
of
papers.
(iii)
Just
as
the
equivalence
between
cartesian
and
polar
representations
of
the
classical
Gaussian
integral
is
used
effectively
to
compute
the
value
of
this
Gauss-
ian
integral,
the
relationship
between
the
Frobenius-
and
étale-pictures
will
play
a
central
role
[cf.,
especially,
the
computations
of
[IUTchIII],
§3;
[IUTchIV],
§1]
in
the
theory
of
the
present
series
of
papers.
64
SHINICHI
MOCHIZUKI
Section
2:
Galois-theoretic
Theta
Evaluation
In
the
present
§2,
we
develop
the
theory
of
group-theoretic
algorithms
sur-
rounding
the
Hodge-Arakelov-theoretic
evaluation
of
the
étale
theta
function
on
l-torsion
points.
At
a
more
technical
level,
this
theory
depends
on
a
careful
analysis
of
the
issue
of
conjugate
synchronization
[cf.
Remark
2.6.1]
—
i.e.,
of
synchronizing
conjugates
of
various
copies
of
objects
associated
to
the
absolute
Galois
group
of
the
base
field
that
occur
at
the
evaluation
points
—
as
well
as
on
the
computation,
via
the
theory
of
[IUTchI],
§2,
of
various
conjugacy
indetermi-
nacies
[cf.
Corollaries
2.4,
2.5]
that
arise
from
the
consideration
of
certain
closed
subgroups
of
various
topological
groups.
In
fact,
these
various
technical
issues
arise,
ultimately,
as
a
consequence
of
the
requirement
of
performing
the
Hodge-
Arakelov-theoretic
evaluation
in
question
with
respect
to
a
single
basepoint
[cf.
the
discussions
of
Remark
1.12.4;
[IUTchI],
Remark
6.12.6].
This
Hodge-Arakelov-
theoretic
evaluation
will
play
a
central
role
in
the
theory
developed
in
the
present
series
of
papers.
In
the
present
§2,
we
shall
work
mainly
with
the
local
portion
at
v
∈
V
bad
of
the
various
mathematical
objects
considered
in
[IUTchI],
§3,
§4,
§5,
§6.
In
fact,
however,
many
of
the
constructions
carried
out
in
the
present
§2
will
be
valid
for
strictly
local
data
[as
in
§1],
i.e.,
that
does
not
necessarily
arise
from
global
data
over
a
number
field.
Nevertheless,
in
order
to
keep
the
notation
simple
from
the
point
of
view
of
discussing
the
compatibility
of
the
theory
of
the
present
§2
with
the
theory
of
[IUTchI],
we
shall
carry
out
the
discussion
of
the
present
§2
only
for
the
localized
objects
that
arise
from
the
theory
of
[IUTchI],
§3,
§4,
§5,
§6,
leaving
the
routine
details
of
a
corresponding
purely
local
theory
to
the
interested
reader.
Proposition
2.1.
Write
(Review
of
Certain
Tempered
Coverings)
Let
v
∈
V
bad
.
Π
tp
Ÿ
⏐
v
⏐
−→
Π
tp
Y
v
⏐
⏐
−→
Π
tp
X
=
Π
v
v
⏐
⏐
Π
tp
Ÿ
−→
Π
tp
Y
v
−→
Π
tp
X
v
v
for
the
diagram
of
open
injections
of
topological
groups
arising
from
the
theory
of
[EtTh],
§2
—
where
tp
(a)
Π
tp
X
,
Π
X
are
the
tempered
fundamental
groups
determined
by
the
hy-
v
v
perbolic
[orbi]curves
X
v
,
X
v
of
[IUTchI],
Definition
3.1,
(e);
tp
tp
tp
(b)
Π
tp
Y
⊆
Π
X
,
Π
Y
v
⊆
Π
X
are
the
open
subgroups
corresponding
to
the
v
v
v
tempered
coverings
Y
v
→
X
v
,
Y
v
→
X
v
determined
by
the
objects
“Y
log
”,
“Y
log
”
in
the
discussion
preceding
[EtTh],
Definition
2.7;
(c)
Π
tp
⊆
Π
tp
X
is
the
open
subgroup
determined
by
the
tempered
covering
Ÿ
v
v
Ÿ
v
→
X
v
of
[IUTchI],
Example
3.2,
(ii);
Π
tp
⊆
Π
tp
X
is
the
open
subgroup
Ÿ
v
v
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
65
corresponding
to
the
tempered
covering
Ÿ
v
→
X
v
determined
by
the
object
“
Ÿ
log
”
in
the
discussion
preceding
[EtTh],
Lemma
1.2;
(d)
the
arrows
are
the
natural
inclusions,
and
both
squares
are
cartesian.
Then
this
diagram
may
be
reconstructed
via
a
functorial
group-theoretic
algo-
rithm
[cf.
[EtTh],
Proposition
2.4]
from
the
[temp-slim!
—
cf.,
e.g.,
[SemiAnbd],
Example
3.10]
topological
group
Π
tp
X
.
v
Proof.
The
assertions
of
Proposition
2.1
follow
immediately
from
the
results
of
[EtTh],
[SemiAnbd]
that
are
quoted
in
the
statements
of
these
assertions.
Remark
2.1.1.
In
the
notation
of
Proposition
2.1:
(i)
Recall
that
the
special
fiber
of
any
model
of
Ÿ
v
that
arises
from
a
stable
model
of
X
v
consists
of
a
chain
of
copies
of
the
projective
line
joined
together
at
the
points
“0”,
“∞”
[cf.
the
discussion
preceding
[EtTh],
Proposition
1.1].
The
set
of
irreducible
components
of
this
special
fiber
may
be
thought
of
as
a
torsor
over
the
group
Z.
Moreover,
the
natural
action
of
Gal(
Ÿ
v
/Y
v
)
∼
=
{±1}
on
Ÿ
v
fixes
each
of
the
irreducible
components
of
the
special
fiber
of
Ÿ
v
and
fits
into
an
exact
sequence
1
→
Gal(
Ÿ
v
/Y
v
)
→
Gal(
Ÿ
v
/X
v
)
→
Gal(Y
v
/X
v
)
→
1,
where
Gal(Y
v
/X
v
)
may
be
identified
with
the
subgroup
l
·Z
⊆
Z.
Since
the
degree
l
covering
X
v
→
X
v
is
totally
ramified
at
the
cusps,
it
thus
follows
that
each
of
the
maps
Γ
Ÿ
→
Γ
Y
;
Γ
Ÿ
→
Γ
Y
;
Γ
Ÿ
→
Γ
Ÿ
;
Γ
Y
→
Γ
Y
;
Γ
X
→
Γ
X
on
dual
graphs
associated
to
the
special
fibers
of
stable
models
[where
we
omit
the
various
subscript
“v’s”
in
order
to
simplify
the
notation]
determined
by
the
various
coverings
discussed
in
Proposition
2.1
induces
a
bijection
on
vertices.
(ii)
Let
ι
X
,
ι
X
,
ι
Ÿ
be
as
in
Remark
1.4.1,
where
we
take
“X
k
”
to
be
X
v
.
Write
ι
Ÿ
for
the
automorphism
of
Ÿ
v
induced
by
ι
Ÿ
;
Γ
X
⊆
Γ
X
for
the
unique
connected
subgraph
of
Γ
X
which
is
a
tree
that
is
stabilized
by
ι
X
and
contains
every
vertex
of
Γ
X
;
Γ
•
X
⊆
Γ
X
for
the
unique
connected
subgraph
of
Γ
X
stabilized
by
ι
X
that
contains
precisely
one
vertex
and
no
edges.
Thus,
if
one
thinks
of
the
vertices
of
Γ
X
as
being
labeled
by
elements
∈
{−l
,
−l
+
1,
.
.
.
,
−1,
0,
1,
.
.
.
,
l
−
1,
l
}
—
where
the
vertex
labeled
0
is
fixed
by
ι
X
—
then
Γ
X
is
obtained
from
Γ
X
by
eliminating
the
unique
edge
joining
the
vertices
with
labels
±l
;
Γ
•
X
consists
of
the
unique
vertex
0
and
no
edges.
In
particular,
by
taking
appropriate
connected
66
SHINICHI
MOCHIZUKI
components
of
inverse
images,
one
concludes
[cf.
(i)]
that
Γ
X
determines
finite,
connected
subgraphs
Γ
•
X
⊆
Γ
X
⊆
Γ
X
,
Γ
•
Ÿ
⊆
Γ
⊆
Γ
Ÿ
,
Ÿ
Γ
•
Ÿ
⊆
Γ
⊆
Γ
Ÿ
Ÿ
of
the
dual
graphs
corresponding
to
X
v
,
Ÿ
v
,
Ÿ
v
which
are
stabilized
by
the
respec-
tive
“inversion
automorphisms”
ι
X
,
ι
Ÿ
,
ι
Ÿ
.
Here,
each
subgraph
Γ
•
(−)
consists
of
precisely
one
vertex
and
no
edges,
while
the
set
of
vertices
of
each
subgraph
Γ
(−)
maps
bijectively
to
the
set
of
vertices
of
Γ
.
In
fact,
[although
we
shall
not
use
this
X
fact
in
the
present
series
of
papers]
it
is
not
difficult
to
verify,
by
considering
the
divisibility
at
the
edges
[i.e.,
nodes]
of
the
divisor
of
poles
of
the
theta
function
[cf.
[EtTh],
Proposition
1.4,
(i)],
that
each
subgraph
Γ
(−)
maps
isomorphically
to
Γ
X
.
Proposition
2.2.
(Decomposition
Groups
Associated
to
Subgraphs)
In
the
notation
of
Proposition
2.1,
write
Π
v•
⊆
Π
v
⊆
Π
v
for
the
decomposition
groups
determined,
respectively,
by
the
subgraphs
Γ
•
X
and
tp
Γ
X
—
i.e.,
more
precisely,
the
group
“Π
X,H
”
of
[IUTchI],
Corollary
2.3,
(iii),
where
we
take
“X”
to
be
X
v
,
“H”
to
be
Γ
•
X
or
Γ
X
,
“Σ”
to
be
{l},
and
“
Σ”
to
be
Primes.
Thus,
Π
v
is
well-defined
up
to
Π
v
-conjugacy;
once
one
fixes
Π
v
,
then
the
subgroup
Π
v•
⊆
Π
v
is
well-defined
up
to
Π
v
-conjugacy
[cf.
Remark
2.2.1
below];
Π
v
⊆
Π
tp
Π
v
=
Π
tp
Y
v
Y
.
Note,
moreover,
that
we
may
assume
that
Π
v•
,
v
def
Π
v
,
and
ι
=
ι
Ÿ
[cf.
Remarks
1.4.1,
(ii);
2.1.1,
(ii)]
have
been
chosen
so
that
some
representative
of
ι
stabilizes
Π
v•
and
Π
v
.
Then:
(i)
The
collection
of
data
(Π
v•
⊆
Π
v
⊆
Π
v
,
ι),
regarded
up
to
Π
v
-conjugacy,
may
be
reconstructed
via
a
functorial
group-theoretic
algorithm
from
the
topo-
logical
group
Π
v
.
(ii)
The
functorial
group-theoretic
algorithms
Π
v
→
θ(Π
v
)
⊆
∞
θ(Π
v
)
⊆
1
lim
−→
H
(Π
Ÿ
(Π
v
)|
J
,
(l
·
Δ
Θ
)(Π
v
))
J
of
Proposition
1.4
[i.e.,
where
we
take
“Π”
to
be
Π
v
],
together
with
the
condition
of
invariance
with
respect
to
ι
[cf.
[EtTh],
Proposition
1.4,
(ii);
the
proof
of
[EtTh],
μ
(Π
v
))-)
Theorem
1.6,
(iii)],
determines
a
specific
μ
2l
-
(respectively,
μ
(=
M
TM
orbit
θ
ι
(Π
v
)
⊆
θ(Π
v
)
(respectively,
ι
∞
θ
(Π
v
)
⊆
∞
θ(Π
v
))
within
the
unique
{(l
·
Z)
×
μ
2l
}-
(respectively,
each
{(l
·
Z)
×
μ}-)
orbit
contained
in
the
set
θ(Π
v
)
(respectively,
∞
θ(Π
v
))
[cf.
Proposition
1.4;
Corollary
1.12,
(ii)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
67
Proof.
Assertion
(i)
follows
immediately
from
the
fact
that
dual
graphs
of
stable
models
may
be
reconstructed
via
a
functorial
group-theoretic
algorithm
from
the
corresponding
tempered
fundamental
group
[cf.,
e.g.,
[SemiAnbd],
Corollary
3.11,
or,
alternatively,
[AbsTopI],
Theorem
2.14,
(i)].
Assertion
(ii)
follows
immediately
from
the
results
of
[EtTh]
that
are
quoted
in
the
statements
of
assertion
(ii).
Remark
2.2.1.
In
the
notation
of
Proposition
2.2,
we
recall
that
since
the
subgroup
Π
v
⊆
Π
v
is
commensurably
terminal
[cf.
[IUTchI],
Corollary
2.3,
(iv)],
it
follows
that
even
when
this
subgroup
is
subject
to
a
Π
v
-conjugacy
indeterminacy,
the
indeterminacy
induced
on
any
specific
Π
v
-conjugate
of
this
subgroup
Π
v
is
an
indeterminacy
with
respect
to
inner
automorphisms
[i.e.,
of
the
specific
Π
v
-
conjugate
of
Π
v
].
Definition
2.3.
(i)
In
the
notation
of
Proposition
2.2;
[IUTchI],
Definition
3.1,
(e);
[IUTchI],
def
def
def
def
def
tp
tp
±
±
cor
cor
=
Δ
tp
=
Remark
3.1.1:
Write
Δ
v
=
Δ
tp
X
,
Δ
v
=
Δ
X
,
Π
v
=
Π
X
,
Δ
v
C
v
,
Π
v
v
v
v
Π
tp
C
v
;
denote
the
respective
profinite
completions
by
means
of
a
“∧”.
Thus,
we
have
natural
diagrams
of
outer
inclusions
of
topological
groups
Δ
v
−→
Δ
±
−→
Δ
cor
Δ
v
−→
Δ
±
v
v
v
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
⏐
Π
v
−→
Π
±
v
−→
Π
cor
v
Π
v
−→
Π
±
v
−→
Δ
cor
v
⏐
⏐
−→
Π
cor
v
—
where
the
left-hand
diagram
admits
a
natural
outer
inclusion
into
the
right-
hand
diagram,
in
the
evident
fashion.
Here,
we
recall
that
Δ
v
includes
as
a
normal
open
subgroup
of
Δ
±
v
of
index
l
[cf.
[EtTh],
Proposition
2.2,
(ii);
[EtTh],
Remark
cor
2.6.1],
that
Δ
±
of
index
2l
[cf.
the
v
includes
as
a
normal
open
subgroup
of
Δ
v
±
discussion
preceding
[EtTh],
Definition
2.1],
and
that
Π
v
and
Π
cor
may
be
recon-
v
structed
group-theoretically
from
Π
v
[cf.
[EtTh],
Proposition
2.4].
We
shall
use
these
diagrams
to
regard
the
various
groups
appearing
in
the
diagrams
as
sub-
cor
groups,
well-defined
up
to
Π
cor
v
-conjugacy,
of
Π
v
.
Recall
the
collection
of
data
(Π
v•
⊆
Π
v
⊆
Π
v
,
ι),
well-defined
up
to
Π
v
-conjugacy,
of
Proposition
2.2,
(i).
Write
Π
±
v•
def
=
N
Π
±
(Π
v•
)
v
⊆
Π
±
v
def
=
N
Π
±
(Π
v
)
v
⊆
Π
±
v
[cf.
Remark
2.1.1,
(ii);
[IUTchI],
Corollary
2.3,
(iv)]
—
so
we
have
natural
isomor-
phisms
∼
∼
∼
∼
±
±
±
∼
Π
±
v•
/Π
v•
→
Π
v
/Π
v
→
Π
v
/Π
v
→
Δ
v
/
Δ
v
→
Gal(X
v
/X
v
)
(
=
Z/lZ)
and
equalities
Π
±
v•
Π
v
=
Π
v•
,
Π
±
v
Π
v
=
Π
v
[cf.
[IUTchI],
Corollary
2.3,
(iv)].
cor
±
cor
(ii)
Let
Π
⊇
,
Π
⊆
be
any
of
the
topological
groups
Π
v
,
Π
±
v
,
Π
v
,
Π
v
,
Π
v
,
Π
v
of
(i);
suppose
that
Π
⊆
⊆
Π
⊇
relative
to
one
of
the
natural
outer
inclusions
discussed
68
SHINICHI
MOCHIZUKI
in
(i).
Then
we
recall
that
the
cuspidal
inertia
groups
of
Π
⊇
may
be
reconstructed
group-theoretically
from
the
topological
group
Π
⊇
via
the
algorithms
of
[AbsTopI],
Lemma
4.5
[cf.
also
[IUTchI],
Remark
1.2.2,
(ii)];
[AbsTopI],
Proposition
4.10,
(vi),
and
that
the
cuspidal
inertia
groups
of
Π
⊆
may
be
obtained
as
the
intersections
with
Π
⊆
of
those
cuspidal
inertia
groups
of
Π
⊇
that
contain
a
finite
index
subgroup
that
lies
inside
Π
⊆
[cf.
[IUTchI],
Corollary
2.5;
[IUTchI],
Remark
2.5.2],
while
the
cuspidal
inertia
groups
of
Π
⊇
may
be
obtained
as
the
Π
⊇
-conjugates
of
the
commensurators
[or,
alternatively,
the
normalizers]
in
Π
⊇
of
the
cuspidal
inertia
groups
of
Π
⊆
[cf.
[CombGC],
Proposition
1.2,
(ii)].
±
(iii)
Let
Π
⊆
be
any
of
the
topological
groups
Π
v
,
Π
±
v
,
Π
v
,
Π
v
of
(i);
if
Π
⊆
def
±
±
is
equal
to
Π
v
or
Π
±
v
,
then
set
Π
⊇
=
Π
v
;
if
Π
⊆
is
equal
to
Π
v
or
Π
v
,
then
set
def
Π
⊇
=
Π
±
v
.
Thus,
Π
⊆
⊆
Π
⊇
.
Then
[cf.
[IUTchI],
Definition
6.1,
(iii)]
we
define
a
±-label
class
of
cusps
of
Π
⊆
to
be
the
set
of
Π
⊆
-conjugacy
classes
of
cuspidal
inertia
subgroups
of
Π
⊆
whose
commensurators
in
Π
⊇
[cf.
the
discussion
of
(ii)]
determine
a
single
Π
⊇
-conjugacy
class
of
subgroups
in
Π
⊇
.
[Here,
we
remark
in
passing
that
since
the
inclusion
Π
⊆
⊆
Π
⊇
corresponds
to
a
totally
ramified
covering
of
curves,
it
is
not
difficult
to
verify
that
such
a
set
of
Π
⊆
-conjugacy
classes
is,
in
fact,
of
cardinality
one.]
Write
LabCusp
±
(Π
⊆
)
def
for
the
set
of
±-label
classes
of
cusps
of
Π
⊆
.
Thus,
when
Π
⊆
=
Π
v
,
if
we
set
†
D
v
=
B
temp
(Π
⊆
)
0
,
then
the
set
LabCusp
±
(Π
⊆
)
may
be
naturally
identified
with
the
set
LabCusp
±
(
†
D
v
)
of
[IUTchI],
Definition
6.1,
(iii).
In
particular,
LabCusp
±
(Π
v
)
=
LabCusp
±
(
†
D
v
)
admits
a
natural
action
by
F
×
l
,
as
well
as
a
zero
element
†
0
η
v
∈
LabCusp
±
(Π
v
)
=
LabCusp
±
(
†
D
v
)
and
a
±-canonical
element
†
±
η
v
∈
LabCusp
±
(Π
v
)
=
LabCusp
±
(
†
D
v
)
—
well-defined
up
to
multiplication
by
±1,
which
may
be
constructed
solely
from
D
v
[cf.
[IUTchI],
Definition
6.1,
(iii)].
†
(iv)
Let
t
∈
LabCusp
±
(Π
v
).
Then
t
determines
a
unique
vertex
of
Γ
X
[cf.
[CombGC],
Proposition
1.5,
(i)].
Write
Γ
•t
X
⊆
Γ
X
for
the
connected
subgraph
with
no
edges
whose
unique
vertex
is
the
vertex
determined
by
t.
Then
just
as
in
the
case
of
Γ
•
X
[i.e.,
the
case
where
t
is
the
zero
element]
discussed
in
Proposition
2.2,
the
subgraph
Γ
•t
X
determines
—
via
a
functorial
group-theoretic
algorithm
—
a
decomposition
group
Π
v•t
⊆
Π
v
⊆
Π
v
—
which
is
well-defined
up
to
Π
v
-conjugacy.
Finally,
we
shall
write
Π
±
v•t
=
∼
(Π
v•t
)
[cf.
(i)];
thus,
we
have
a
natural
isomorphism
Π
±
N
Π
±
v•t
/Π
v•t
→
Gal(X
v
/X
v
).
v
def
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
69
±
±
(v)
Let
Π
⊆
be
either
of
the
topological
groups
Π
±
v
,
Π
v
of
(i);
if
Π
⊆
=
Π
v
,
then
def
def
±
cor
set
Π
⊇
=
Π
cor
v
;
if
Π
⊆
=
Π
v
,
then
set
Π
⊇
=
Π
v
.
Then
one
verifies
immediately
that
the
images
[via
the
natural
outer
injection
Π
v
→
Π
⊆
]
in
LabCusp
±
(Π
⊆
)
of
the
various
structures
on
LabCusp
±
(Π
v
)
reviewed
in
(iii)
determine
[in
the
notation
and
terminology
of
[IUTchI],
Definition
6.1,
(i)]
a
natural
F
±
l
-torsor
structure
on
±
LabCusp
(Π
⊆
).
Moreover,
the
natural
action
of
Π
⊇
/Π
⊆
on
Π
⊆
preserves
this
F
±
l
-torsor
structure,
hence
determines
a
natural
outer
isomorphism
∼
Π
⊇
/Π
⊆
→
F
±
l
[cf.
[IUTchI],
Definition
6.1,
(i)].
Remark
2.3.1.
In
the
situation
of
(iii),
suppose
that
the
inclusion
Π
⊆
⊆
Π
⊇
is
strict.
Then
one
verifies
immediately
that
if
I
⊆
Π
⊇
is
a
cuspidal
inertia
group
of
Π
⊇
,
then
the
cuspidal
inertia
group
I
Π
⊆
⊆
Π
⊆
of
Π
⊆
satisfies
I
Π
⊆
=
I
l
—
where
the
superscript
l
is
relative
to
the
group
operation
on
I,
written
multi-
plicatively.
In
particular,
[even
though
Π
v
(respectively,
Π
v
)
fails
to
be
normal
in
cor
±
±
cor
Π
cor
v
(respectively,
Π
v
)]
it
follows
—
since
Π
v
(respectively,
Π
v
)
is
normal
in
Π
v
(respectively,
Π
cor
v
)
—
that
the
cuspidal
inertia
groups
of
Π
v
(respectively,
Π
v
)
are
permuted
by
the
conjugation
action
of
Π
cor
(respectively,
Π
cor
v
v
).
The
theta
evaluation
algorithm
discussed
in
the
following
Corollaries
2.4,
2.5,
2.8,
and
2.9
is
central
to
the
theory
of
the
present
§2,
and,
indeed,
of
the
present
series
of
papers.
Corollary
2.4.
(F
±
l
-Symmetric
Two-torsion
Translates
of
Cusps)
In
the
notation
of
Definition
2.3:
Let
t
∈
LabCusp
±
(Π
v
);
∈
{•t,
}.
Write
def
def
±
Π
v
,
Δ
±
Π
±
Δ
v
=
Δ
v
=
Δ
v
v
v
def
Π
v
¨
=
Π
v
Π
tp
,
Ÿ
v
def
Δ
v
¨
=
Δ
v
Π
v
¨
—
so
we
have
[Π
v
:
Π
v
¨
]
=
[Δ
v
:
Δ
v
¨
]
=
2,
±
[Π
±
v
:
Π
v
]
=
[Δ
v
:
Δ
v
]
=
l
±
[Π
±
¨
]
=
[Δ
v
:
Δ
v
¨
]
=
2l
v
:
Π
v
[cf.
Definition
2.3,
(i),
(iv)].
(i)
(Inclusions
and
Conjugates)
Let
I
t
⊆
Π
v
be
a
cuspidal
inertia
group
that
belongs
to
the
class
determined
by
t
such
that
I
t
⊆
Δ
v
.
Consider
the
[
Π
±
v
-
conjugacy
stable]
sets
of
subgroups
of
Π
±
v
{I
t
γ
1
}
γ
∈
Π
±
=
{I
t
γ
1
}
γ
∈
Δ
±
1
v
1
v
70
SHINICHI
MOCHIZUKI
2
2
{Π
γ
v
}
γ
∈
Π
±
=
{Π
γ
v
}
γ
∈
Δ
±
;
2
v
2
±
γ
3
γ
3
{(Π
±
v
)
}
γ
∈
Π
±
=
{(Π
v
)
}
γ
∈
Δ
±
v
3
v
3
v
—
where
the
superscript
“γ’s”
denotes
conjugation
[i.e.,
“(−)
→
γ
·
(−)
·
γ
−1
”]
by
γ.
Then
for
γ,
γ
∈
Δ
±
v
,
the
following
three
conditions
are
equivalent:
(a)
γ
∈
Δ
±
v
;
(b)
I
t
γ·γ
⊆
Π
γv
;
γ
(c)
I
t
γ·γ
⊆
(Π
±
v
)
.
(ii)
(Two-torsion
Translates
of
Cusps)
In
the
situation
of
(i),
if
we
write
def
δ
=
γ
·
γ
∈
Δ
±
v
,
then
any
inclusion
I
t
δ
=
I
t
γ·γ
⊆
Π
γv
=
Π
δv
as
in
(i)
completely
determines
the
following
data:
def
(a)
a
decomposition
group
D
t
δ
=
N
Π
δv
(I
t
δ
)
⊆
Π
δv
¨
corresponding
to
the
inertia
group
I
t
δ
;
±
δ
(b)
a
decomposition
group
D
μ
δ
−
⊆
Π
δv
¨
,
well-defined
up
to
(Π
v
)
-
[or,
δ
equivalently,
(Δ
±
v
)
-]
conjugacy,
corresponding
to
the
torsion
point
“μ
−
”
of
Remark
1.4.1,
(i),
(ii),
via
the
algorithms
of
[SemiAnbd],
Theorem
6.8,
(iii)
[concerning
the
group-theoreticity
of
the
decomposition
groups
of
torsion
points],
and
[SemiAnbd],
Corollary
3.11
[concerning
the
dual
semi-graphs
of
the
special
fibers
of
stable
models],
applied
to
Δ
δv
⊆
Π
δv
;
±
δ
δ
(c)
a
decomposition
group
D
t,μ
⊆
Π
δv
¨
,
well-defined
up
to
(Π
v
)
-
−
δ
[or,
equivalently,
(Δ
±
v
)
-]
conjugacy
—
i.e.,
the
image
of
an
evaluation
section
[cf.
[IUTchI],
Example
4.4,
(i)]
—
corresponding
to
the
“μ
−
-
translate
of
the
cusp
that
gives
rise
to
I
t
δ
”,
via
the
algorithm
of
[SemiAnbd],
Theorem
6.8,
(iii)
[concerning
the
group-theoreticity
of
the
decomposition
groups
of
translates
by
torsion
points
of
the
cusps].
Moreover,
the
construction
of
the
above
data
is
compatible
with
conjugation
by
arbitrary
δ
∈
Δ
±
v
,
as
well
as
with
the
natural
inclusion
Π
v•t
⊆
Π
v
of
Definition
2.3,
(iv),
as
one
varies
∈
{•t,
}.
(iii)
(F
±
l
-Symmetry)
Suppose
that
=
•t.
Then
the
construction
of
the
[cf.
data
of
(ii),
(a),
(c),
is
compatible
with
conjugation
by
arbitrary
δ
∈
Π
cor
v
Remark
2.3.1].
Here,
we
recall
from
Definition
2.3,
(v),
that
we
have
natural
outer
±
±
∼
±
∼
Π
cor
isomorphisms
Δ
cor
v
/
Δ
v
→
v
/
Π
v
→
F
l
.
Proof.
First,
we
consider
assertion
(i).
The
implications
(a)
=⇒
(b)
and
(b)
=⇒
(c)
are
immediate
from
the
definitions
[cf.
also
Remark
2.3.1].
Thus,
it
suffices
to
±
γ
verify
that
(c)
=⇒
(a),
i.e.,
that
the
condition
I
t
γ·γ
⊆
(Π
±
v
)
implies
that
γ
∈
Δ
v
;
we
may
assume
without
loss
of
generality
that
γ
=
1.
Then
by
[IUTchI],
Corollary
±
2.5
[cf.
also
[IUTchI],
Remark
2.5.2],
the
inclusion
I
t
γ
⊆
Π
±
v
⊆
Π
v
implies
that
γ
∈
Δ
±
v
.
Now,
by
applying
the
equivalence
of
[IUTchI],
Corollary
2.3,
(vi)
[cf.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
71
also
[CombGC],
Proposition
1.2,
(ii)],
to
the
various
finite
index
open
subgroups
±
of
Δ
±
v
,
it
follows
that
γ
∈
Δ
v
—
where
we
use
the
notation
“∧”
to
denote
the
closure
in
Δ
±
v
[cf.
Proposition
2.2;
Definition
2.3,
(iv);
[IUTchI],
Corollary
2.3,
±
(ii)]
—
hence
that
γ
∈
Δ
±
Δ
±
v
[cf.
[IUTchI],
Corollary
2.3,
(v)].
This
v
=
Δ
v
completes
the
proof
of
assertion
(i).
Assertions
(ii)
and
(iii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.4.1.
Note
that
by
applying
[IUTchI],
Proposition
2.4,
(i)
[cf.
the
proof
of
[IUTchI],
Corollary
2.5;
[IUTchI],
Remark
2.5.2],
one
may
replace
“I
t
”
in
Corollary
2.4
by
its
maximal
pro-l
subgroup
for
any
l
∈
Primes
\
{p
v
}.
The
use
of
such
maximal
pro-l
subgroups
sometimes
results
in
a
simplification
of
arguments
involving
intersections
with
other
closed
subgroups,
since
every
closed
subgroup
of
such
a
maximal
pro-l
subgroup
is
either
open
or
trivial.
(Group-theoretic
Theta
Evaluation)
In
the
notation
of
Corollary
2.5.
Corollary
2.4:
(i)
(Restriction
of
Subquotients
to
Subgraphs)
Write
(l
·
Δ
Θ
)(Π
v
¨
)
for
the
subquotient
of
Π
v
¨
determined
by
the
subquotient
(l
·
Δ
Θ
)(Π
v
)
of
Π
v
.
Then
∼
the
inclusion
Π
v
¨
→
Π
v
induces
an
isomorphism
(l
·
Δ
Θ
)(Π
v
¨
)
→
(l
·
Δ
Θ
)(Π
v
).
Write
Π
v
G
v
(Π
v
),
Π
v
¨
G
v
(Π
v
¨
)
for
the
quotients
determined
by
the
natural
surjection
Π
v
G
v
.
Then
there
exists
a
functorial
group-theoretic
algorithm
for
constructing
these
quotients
from
the
topological
group
Π
v
[cf.,
e.g.,
[AbsAnab],
Lemma
1.3.8,
as
well
as
Proposition
2.2,
(i);
Corollary
2.4
of
the
present
paper].
(ii)
(Restriction
of
Étale
Theta
Functions
to
Subgraphs
and
Evalua-
tion
Points)
Let
γ
δ
I
t
δ
=
I
t
γ·γ
⊆
Π
δv
¨
⊆
Π
v
=
Π
v
def
be
an
inclusion
as
in
Corollary
2.4,
(ii)
[where
we
take
=
].
Then
restriction
of
the
ι
γ
-invariant
sets
θ
ι
(Π
γv
),
∞
θ
ι
(Π
γv
)
of
Proposition
2.2,
(ii),
to
the
subgroup
γ
γ
Π
γv
¨
⊆
Π
Ÿ
(Π
v
)
(⊆
Π
v
)
yields
μ
2l
-,
μ-orbits
of
elements
θ
ι
(Π
γv
¨
)
⊆
γ
ι
∞
θ
(Π
v
¨
)
γ
γ
1
lim
¨
|
J
,
(l
·
Δ
Θ
)(Π
v
¨
))
−→
H
(Π
v
⊆
J
—
where
J
⊆
Π
v
ranges
over
the
open
subgroups
of
Π
v
—
which,
upon
further
δ
of
Corollary
2.4,
(ii),
(c),
yield
restriction
to
the
decomposition
groups
D
t,μ
−
μ
2l
-,
μ-orbits
of
elements
θ
t
(Π
γv
¨
)
⊆
γ
t
∞
θ
(Π
v
¨
)
⊆
γ
γ
1
lim
¨
)|
J
G
,
(l
·
Δ
Θ
)(Π
v
¨
))
−→
H
(G
v
(Π
v
J
G
72
SHINICHI
MOCHIZUKI
∼
for
each
t
∈
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
—
where
J
G
⊆
G
v
(Π
γv
¨
)
ranges
over
∼
the
open
subgroups
of
G
v
(Π
γv
¨
);
the
“
→
”
is
induced
by
conjugation
by
γ.
Moreover,
γ
t
the
sets
θ
t
(Π
γv
¨
),
∞
θ
(Π
v
¨
)
depend
only
on
the
label
|t|
∈
|F
l
|
determined
by
t
[cf.
Definition
2.3,
(iii);
[IUTchI],
Example
4.4,
(i);
[IUTchI],
Definition
6.1,
(iii)].
γ
γ
γ
t
|t|
t
Thus,
we
shall
write
θ
|t|
(Π
γv
¨
)
=
θ
(Π
v
¨
),
∞
θ
(Π
v
¨
)
=
∞
θ
(Π
v
¨
).
def
def
(iii)
(Functorial
Group-theoretic
Evaluation
Algorithm)
If
one
starts
γ
with
an
arbitrary
Δ
±
¨
,
and
one
considers,
as
t
ranges
v
-conjugate
Π
v
¨
of
Π
v
∼
∼
over
the
elements
of
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
[where
the
“
→
”
is
induced
γ
|t|
by
conjugation
by
γ],
the
resulting
μ
2l
-,
μ-orbits
θ
|t|
(Π
γv
¨
),
∞
θ
(Π
v
¨
)
arising
from
γ
δ
an
arbitrary
Δ
±
v
-conjugate
I
t
of
I
t
that
is
contained
in
Π
v
¨
[cf.
(ii)],
then
one
obtains
a
group-theoretic
algorithm
for
constructing
the
collections
of
μ
2l
-,
μ-
orbits
{
∞
θ
|t|
(Π
γv
{θ
|t|
(Π
γv
¨
)}
|t|∈|F
l
|
;
¨
)}
|t|∈|F
l
|
which
is
functorial
in
the
topological
group
Π
v
and,
moreover,
compatible
with
γ
1
γ
1
the
independent
conjugacy
actions
of
Δ
±
v
on
the
sets
{I
t
}
γ
∈
Π
±
=
{I
t
}
γ
∈
Δ
±
1
v
1
v
and
{Π
γ
v
2
=
{Π
γ
v
2
[cf.
the
sets
of
Corollary
2.4,
(i);
Remark
2.2.1].
¨
}
γ
2
∈
Π
±
¨
}
γ
2
∈
Δ
±
v
v
Proof.
Assertions
(i),
(ii),
and
(iii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Here,
in
assertion
(i),
we
observe
that
the
fact
that
the
inclusion
Π
v
¨
→
Π
v
induces
an
isomorphism
∼
(l·Δ
Θ
)(Π
v
¨
)
→
(l·Δ
Θ
)(Π
v
)
follows
immediately
by
considering
the
cuspidal
inertia
groups
involved.
Remark
2.5.1.
(i)
Recall
from
the
discussion
of
[IUTchI],
Example
4.4,
(i),
that
relative
to
the
“standard”
cyclotomic
rigidity
isomorphism
(∗
bs-Gal
)
of
Proposition
1.3,
(ii),
and
the
resulting
Kummer
map
K
v
×
→
H
1
(G
v
(Π
v
¨
),
(l
·
Δ
Θ
)(Π
v
¨
))
[i.e.,
we
take
“δ”
in
Corollary
2.5,
(ii),
to
be
the
identity
—
without
loss
of
generality,
in
light
of
Remark
2.2.1],
it
follows
immediately
from
the
definition
of
the
connected
subgraph
“Γ
X
”
in
Remark
2.1.1,
(ii)
[cf.
also
[IUTchI],
Corollary
2.3,
(vi)],
that,
for
j
∈
|F
l
|,
the
set
θ
j
(Π
v
¨
)
consists
of
precisely
the
μ
2l
-orbit
of
the
“theta
value”
q
j
2
v
[cf.
[IUTchI],
Example
3.2,
(iv);
[EtTh],
Proposition
1.4,
(ii)]
—
where
the
“j”
in
the
exponent
denotes
the
element
∈
{0,
1,
.
.
.
,
l
}
determined
by
the
given
element
j
∈
|F
l
|.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
73
(ii)
Note
that
[the
reciprocals
of
the
l-th
powers
of]
the
theta
values
discussed
in
(i)
are
somewhat
unusual
among
the
various
values
Θ̈(c)
—
where
c
∈
K
v
—
attained
by
the
theta
series
def
−
1
Θ̈
=
Θ̈(
Ü
)
=
q
X
8
·
n∈Z
1
(n+
12
)
2
(−1)
n
·
q
X
2
·
Ü
2n+1
discussed
in
[EtTh],
Proposition
1.4
[cf.
the
notation
of
loc.
cit.]
in
that
they
satisfy
the
following
crucial
property
[cf.
the
discussion
of
Remark
1.12.2]:
the
ratio
Θ̈(c)/
Θ̈(c
)
is
a
root
of
unity,
for
any
c
∈
K
v
[corresponding
to
a
point
of
Ÿ
v
]
that
occurs
as
the
result
of
applying
an
automorphism
of
Π
v
to
[the
point
of
Ÿ
v
that
corresponds
to]
c
such
that
c
/c
is
a
unit.
That
is
to
say,
the
reciprocals
of
the
l-th
powers
of
the
theta
values
discussed
in
(i)
j/2
√
correspond
to
the
values
Θ̈(±
−1
·
q
X
),
where
j
∈
{0,
1,
.
.
.
,
l
},
i.e.,
the
values
j/2
√
at
points
separated
by
periods
[i.e.,
the
“q
X
”]
from
the
point
“±
−1”.
These
values
may
be
computed
easily
from
the
“functional
equations”
given
in
[EtTh],
Proposition
1.4,
(ii).
(iii)
Note
that,
in
the
context
of
the
F
±
l
-symmetry
discussed
in
Corollary
2.4,
(iii),
the
various
μ
2l
-multiple
indeterminacies
that
occur,
for
various
j
∈
|F
l
|,
in
the
μ
2l
-orbit
θ
j
(Π
v
¨
)
are
independent.
That
is
to
say,
these
indeterminacies
are
not
“synchronized”
so
as
to
arise
from
a
single
indeterminacy
that
is
independent
of
j.
Indeed,
each
of
these
μ
2l
-multiple
δ
indeterminacies
arises
as
a
consequence
of
the
action
of
(Δ
±
•
t
)
,
where
we
v•t
/Δ
v¨
recall
from
Corollary
2.4
that
[Δ
±
•
t
]
=
2l,
on
the
decomposition
groups
v•t
:
Δ
v¨
δ
±
“D
t,μ
⊆
Π
δv
¨
”
of
Corollary
2.4,
(ii),
(c),
hence
is
induced
by
the
Δ
v
-outer
nature
−
∼
±
±
of
the
action
of
Δ
cor
that
appears
in
Corollary
2.4,
(iii)
—
cf.
the
v
/
Δ
v
→
F
l
discussion
of
Remarks
2.5.2,
2.6.2
below.
Remark
2.5.2.
(i)
If
one
thinks
of
the
“set
{I
t
γ
1
}
γ
∈
Π
±
=
{I
t
γ
1
}
γ
∈
Δ
±
regarded
up
to
Δ
±
v
-conjugacy”
1
v
1
v
(respectively,
“set
{Π
γ
v
2
=
{Π
γ
v
2
regarded
up
to
Δ
±
v
-conjugacy”)
¨
}
γ
2
∈
Π
±
¨
}
γ
2
∈
Δ
±
v
v
[cf.
Corollary
2.5,
(iii)]
as
a
sort
of
quotient
by
Δ
±
v
,
then
one
may
think
of
the
γ
1
γ
2
various
inclusion
morphisms
I
t
→
Π
v
¨
as
a
sort
of
morphism
between
quotients
γ
1
Δ
±
v
{I
t
}
γ
1
∈
Δ
±
v
/
Δ
±
v
→
γ
2
Δ
±
v
{Π
v
¨
}
γ
2
∈
Δ
±
v
/
Δ
±
v
74
SHINICHI
MOCHIZUKI
which
induces
a
morphism
between
quotients
γ
1
γ
2
±
±
/
Δ
/
Δ
±
Δ
±
{D
}
→
Δ
{Π
}
±
±
t,μ
−
γ
∈
Δ
v
v
v
v
¨
γ
∈
Δ
v
1
v
v
2
—
cf.
Corollary
2.4,
(ii);
the
discussion
of
[IUTchI],
Remark
4.5.1,
(i),
(iii).
Since
all
of
the
inclusions
involved
occur
within
a
single
“ambient
container”
—
namely,
±
Π
±
v
,
regarded
up
to
Π
v
-conjugacy
—
the
evaluation
algorithm
discussed
in
Corollary
2.5,
(iii),
may
be
thought
of
as
a
sort
of
“nested”
version
of
the
principle
of
“Galois
evaluation”
discussed
in
Remark
1.12.4.
Here,
we
note
that
unlike
the
situation
discussed
in
Remark
1.12.4,
in
which
the
subgroup
Π
Ÿ
(Π)
⊆
Π
is
normal,
±
the
subgroups
Π
v
,
Π
v
¨
⊆
Π
v
are
far
from
being
normal!
(ii)
In
the
notation
of
[IUTchI],
Definition
3.1,
(d)
[cf.
also
the
notation
of
[IUTchI],
Definition
6.1,
(v)],
write
Π
±
=
Π
X
K
;
def
Δ
±
=
Δ
X
def
—
so
Δ
±
may
be
naturally
identified,
up
to
inner
automorphism,
with
Δ
±
v
.
Then
±
note
that
unlike
the
tempered
fundamental
groups
Δ
v
,
Δ
v
,
Δ
v
,
Δ
v
¨
or
the
local
±
±
Galois
groups
Π
v
/Δ
v
,
Π
v
/Δ
v
,
Π
v
/Δ
v
,
Π
v
¨
/Δ
v
¨
—
all
of
which
depend,
in
a
quite
essential
way,
on
v
—
the
topological
group
Δ
±
∼
=
Δ
±
is
independent
of
v
v
and,
moreover,
may
be
recovered
directly
from
the
global
portion
“
†
D
±
”
of
a
D-Θ
ell
-bridge
[cf.
[IUTchI],
Definition
6.4,
(ii);
[AbsAnab],
Lemma
1.1.4,
(i)].
On
the
other
hand,
Δ
±
∼
=
Δ
±
v
also
serves
as
an
“ambient
container”
for
the
Δ
±
¨
.
That
is
to
say,
v
-conjugates
of
both
I
t
and
Δ
v
Δ
±
(
∼
=
Δ
±
v
)
serves
as
a
sort
of
“common
bridge”
between
local
data
[such
as
Δ
v
¨
]
and
global
data
such
as
the
labels
t
∈
LabCusp
±
(Π
±
)
∼
∼
(
→
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
))
[where
we
write
LabCusp
±
(Π
±
)
=
LabCusp
±
(B(Π
±
)
0
)
—
cf.
[IUTchI],
Definition
6.1,
(vi)],
in
the
form
of
conjugacy
classes
of
I
t
.
def
(iii)
On
the
other
hand,
if,
in
the
discussion
of
(ii),
one
passes
—
as
in
the
theory
of
[IUTchI],
§6
—
between
distinct
v
via
this
“global
bridge”
Δ
±
,
then
one
must
take
into
account
the
fact
that,
unlike
the
labels
t
[i.e.,
conjugacy
classes
of
I
t
],
the
groups
Π
v
¨
do
not
admit
globalizations
or
extensions
to
multiple
v’s.
This
is
precisely
the
reason
for
∼
±
)-
[or,
equivalently,
Π
±
-]
conjugacy
the
independence
of
the
Δ
±
v
(
=
Δ
v
indeterminacies
that
act
on
the
conjugates
of
I
t
and
Π
v
¨
[cf.
the
“quotient
interpretation”
of
(i)
above;
the
statement
of
Corollary
2.5,
(iii)].
Here,
we
observe
that
since
[in
the
notation
of
[IUTchI],
Definition
3.1]
neither
of
±
G
K
admits
a
section
that
simultaneously
the
natural
surjections
Π
±
v
G
v
,
Π
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
75
∼
normalizes
the
subgroups
I
t
,
as
t
ranges
over
the
elements
of
LabCusp
±
(Π
±
)
→
∼
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
[cf.,
e.g.,
[AbsSect],
Theorem
1.3,
(ii);
[pGC],
Theorem
C],
it
follows
that
any
G
v
-
(respectively,
G
K
-)
conjugacy
indeterminacy
∼
±
-conjugacy
indeterminacy
acting
on
the
various
necessarily
results
in
a
Δ
±
v
=
Δ
I
t
,
i.e.,
G
v
-conjugacy
indeterminacy
=⇒
Δ
±
v
-conjugacy
indeterminacy,
G
K
-conjugacy
indeterminacy
=⇒
Δ
±
-conjugacy
indeterminacy.
cor
±
Since,
moreover,
the
natural
surjection
Δ
cor
v
Δ
v
/
Δ
v
does
not
admit
a
splitting,
∼
±
cor
±
it
follows
that
the
Δ
±
of
Corollary
2.4,
(iii),
v
-outer
action
of
Δ
v
/
Δ
v
→
F
l
induces
∼
±
-conjugacy
indeterminacies
on
the
subgroups
I
t
,
independent
Δ
±
v
=Δ
for
distinct
t.
In
a
similar
vein,
since
G
v
does
not
determine
a
direct
summand
of
G
K
—
cf.
[NSW],
Corollary
12.1.3;
the
phenomenon
of
the
non-splitting
of
“prime
decomposition
trees”
discussed
in
[IUTchI],
Remark
4.3.1,
(ii)
—
it
follows
that
any
G
K
-conjugacy
indeterminacy
[which,
as
just
discussed,
gives
rise
to
Δ
±
-conjugacy
indeterminacy]
induces
independent
G
v
-conjugacy
indeterminacies
on
the
various
G
K
-conjugates
of
G
v
[hence
also,
as
just
discussed,
independent
Δ
±
v
-conjugacy
indeterminacies]
—
i.e.,
G
K
-conjugacy
indeterminacy
=⇒
independent
G
v
-conjugacy
indeterminacies
—
cf.
the
discussion
of
[IUTchI],
Remark
4.5.1,
(iii).
(iv)
One
way
to
visualize
the
independent
conjugacy
indeterminacies
of
the
discussion
of
(iii)
above
is
via
the
illustration
given
in
Fig.
2.1
below.
...
◦
◦
◦
◦
◦
...
...
•
−→
•
−→
•
−→
•
−→
•
...
Fig.
2.1:
Independent
conjugacy
indeterminacies
That
is
to
say,
one
thinks
of
the
upper
and
lower
lines
of
Fig.
2.1
as
being
equipped
with
independent
actions
by
groups
of
horizontal
translations
[i.e.,
each
of
which
is
isomorphic
to
Z];
one
thinks
of
each
of
the
“◦’s”
in
the
upper
line
as
representing
a
Δ
±
∼
=
Δ
±
v
-conjugate
of
I
t
and
of
each
of
the
“•
−→
•’s”
in
the
lower
line
as
representing
a
Δ
±
∼
=
Δ
±
-conjugate
of
Π
v
¨
.
Thus,
since
the
translation
actions
on
v
the
upper
and
lower
lines
are
not
synchronized
with
one
another
[cf.
the
discussion
of
(iii)],
there
is
no
way
to
separate
—
i.e.,
in
a
fashion
that
is
compatible
with
the
indeterminacy
arising
from
both
translation
actions
—
the
inclusion
of
a
“◦”
into
a
“•
−→
•”
as
the
left-hand
“•”
from
the
inclusion
of
the
same
“◦”
into
some
“•
−→
•”
as
the
right-hand
“•”.
76
SHINICHI
MOCHIZUKI
Corollary
2.6.
lary
2.5,
(ii):
(Splittings
Defined
on
Subgraphs)
In
the
notation
of
Corol-
×
(i)
(“M
TM
”
Defined
on
Subgraphs)
The
γ-conjugate
of
the
quotient
Π
v
¨
G
v
(Π
v
¨
)
of
Corollary
2.5,
(i),
determines
subsets
γ
γ
γ
×
1
1
H
(J
,
(l
·
Δ
)(Π
))
⊇
M
TM
(Π
γv
lim
G
Θ
¨
¨
)
⊆
lim
¨
|
J
,
(l
·
Δ
Θ
)(Π
v
¨
)),
−→
H
(Π
v
−→
v
J
G
×
M
TM
·
θ
ι
(Π
γv
¨
)
⊆
×
M
TM
·
∞
θ
ι
(Π
γv
¨
)
J
γ
γ
1
⊆
lim
¨
|
J
,
(l
·
Δ
Θ
)(Π
v
¨
))
−→
H
(Π
v
J
—
where
J
G
⊆
G
v
(Π
v
¨
),
J
⊆
Π
v
range
over
the
open
subgroups
of
G
v
(Π
v
¨
),
Π
v
,
×
×
×
×
respectively;
M
TM
·
θ
ι
(−)
=
M
TM
(−)
·
θ
ι
(−),
M
TM
·
∞
θ
ι
(−)
=
M
TM
(−)
·
∞
θ
ι
(−)
—
which
are
compatible,
relative
to
the
first
restriction
operation
discussed
×
×
(−)”,
“M
TM
·
θ
ι
(−)”,
in
Corollary
2.5,
(ii),
with
the
corresponding
subsets
“M
TM
×
“M
TM
·
∞
θ
ι
(−)”
of
Proposition
1.4
and
Corollary
1.12
[cf.
Corollary
1.12,
(a),
(c),
(e);
Corollary
1.12,
(i);
Remark
1.11.5,
(i),
(ii)].
Also,
[cf.
Corollary
1.12]
let
us
write
def
×μ
γ
μ
γ
×
(Π
γv
M
TM
¨
)
=
M
TM
(Π
v
¨
)/M
TM
(Π
v
¨
)
def
def
μ
γ
×
—
where
M
TM
(Π
γv
¨
)
⊆
M
TM
(Π
v
¨
)
denotes
the
submodule
of
torsion
elements.
(ii)
(Splittings
at
Zero-labeled
Evaluation
Points)
In
the
situation
of
Corollary
2.5,
(ii),
suppose
that
t
is
taken
to
be
the
zero
element.
Then
the
γ
t
set
θ
t
(Π
γv
¨
)
(respectively,
∞
θ
(Π
v
¨
))
is
equal
to
the
μ
2l
-
(respectively,
μ-)
orbit
of
the
identity
element
[i.e.,
the
zero
element
of
cohomology
module
in
question,
if
one
denotes
the
module
structure
additively].
In
particular,
if
one
considers
the
μ
(Π
γv
quotient
of
the
diagram
of
the
first
display
of
(i)
by
M
TM
¨
),
then
restriction
δ
of
Corollary
2.4,
(ii),
(c),
determines
splittings
to
the
decomposition
groups
D
t,μ
−
×μ
γ
μ
γ
ι
M
TM
(Π
γv
¨
)
×
{
∞
θ
(Π
v
¨
)/M
TM
(Π
v
¨
)}
μ
γ
×
·
∞
θ
ι
(Π
γv
of
M
TM
¨
)/M
TM
(Π
v
¨
)
which
are
compatible,
relative
to
the
first
restric-
tion
operation
discussed
in
Corollary
2.5,
(ii),
with
the
splittings
of
Corollary
1.12,
(ii).
Proof.
Assertions
(i)
and
(ii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.6.1.
(i)
One
of
the
most
central
properties,
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
of
the
evaluation
algorithm
of
Corollary
2.5,
(iii),
consists
of
the
observation
that
this
algorithm
is
performed
relative
to
a
single
basepoint
—
i.e.,
from
a
more
geometric
point
of
view,
relative
to
the
“fundamental
group”
Π
γv
¨
corresponding
to
the
connected
subgraph
Γ
X
⊆
Γ
X
[cf.
Remark
2.1.1,
(ii)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
77
In
particular,
despite
the
fact
that
we
are
ultimately
interested
in
[not
a
single,
but
rather]
a
plurality
of
theta
values,
associated
to
the
various
|t|
∈
|F
l
|,
these
theta
values
γ
θ
|t|
(Π
γv
⊆
H
1
(G
v
(Π
γv
¨
)
¨
),
(l
·
Δ
Θ
)(Π
v
¨
))
for
various
|t|
∈
|F
l
|
are
all
computed
relative
to
the
single
copy
[i.e.,
which
is
independent
of
|t|!]
of
the
Galois
group
G
v
(Π
γv
¨
)
and
the
single
cyclotome
γ
γ
(l
·
Δ
Θ
)(Π
v
¨
)
[i.e.,
which
is
independent
of
|t|!]
arising
from
Π
v
¨
—
i.e.,
arising
from
the
“single
basepoint”
under
consideration.
We
shall
refer
to
this
phenom-
enon
by
the
term
conjugate
synchronization.
This
conjugate
synchronization
is
necessary
in
order
to
perform
Kummer
theory
[cf.
the
discussion
of
Galois
evaluation
in
Remark
1.12.4],
as
we
shall
do
in
§3.
(ii)
Put
another
way,
the
significance
of
conjugate
synchronization
in
the
con-
text
of
Kummer
theory
—
especially,
in
the
context
of
the
theory
of
Gaussian
Frobenioids,
to
be
developed
in
§3
below
—
may
be
understood
as
arising
from
the
requirement
that
the
collection
of
theta
values,
for
|t|
∈
F
l
,
be
treated
as
a
single
unified
entity,
whose
Kummer
theory
may
be
described
by
considering
the
action
of
a
single
Galois
group
in
the
context
of
the
simultaneous
extraction
of
N
-th
roots
of
all
theta
values,
relative
to
a
single
cyclotome
[i.e.,
copy
of
the
module
of
N
-th
roots
of
unity]
that
acts
simultaneously
on
the
N
-th
roots
of
all
of
the
theta
values,
and
in
a
fashion
that
is
compatible
with
the
Kummer
theory
of
the
“base
field”
γ
[i.e.,
arising
from
the
quotient
Π
γv
¨
G
v
(Π
v
¨
)].
This
point
of
view
may
only
be
realized
by
means
of
a
“single
basepoint”
of
a
suitable
category
of
coverings
of
a
geometric
object
that
consists
of
a
single
connected
component
[cf.
the
discussion
of
Galois
evaluation
in
Remark
1.12.4;
the
discussion
of
[EtTh],
Remark
1.10.4].
Also,
we
recall
[cf.
the
discussion
of
Galois
evaluation
in
Remark
1.12.4]
that
this
“Kummer-theoretic
representation”
of
the
[“Frobenioid-theoretic”]
monoid
generated
by
the
[“Frobenioid-theoretic”]
theta
function
satisfies
the
crucial
property
of
being
compatible
[unlike
the
various
ring
structures
involved!]
with
the
“log-wall”
[cf.
the
theory
of
[AbsTopIII]].
This
crucial
property
will
play
a
fundamental
role
in
the
theory
to
be
developed
in
[IUTchIII].
Remark
2.6.2.
(i)
In
the
context
of
the
discussion
of
conjugate
synchronization
in
Remark
2.6.1,
it
is
useful
to
recall
the
theory
of
D-Θ
±ell
-Hodge
theaters
†
HT
D-Θ
±ell
†
=
(
D
†
±
φ
Θ
±
←−
†
D
T
†
ell
φ
Θ
±
−→
†
D
±
)
[cf.
[IUTchI],
Definition
6.4,
(iii)]
developed
in
[IUTchI],
§6.
That
is
to
say,
from
the
point
of
view
of
the
theory
of
D-Θ
±ell
-Hodge
theaters,
it
is
natural
to
think
(a)
of
the
topological
group
Π
v
that
appears
in
Corollaries
2.4,
2.5,
and
2.6
as
the
tempered
fundamental
group
of
†
D
,v
,
78
SHINICHI
MOCHIZUKI
(b)
of
the
topological
group
Π
±
v
that
appears
in
Corollaries
2.4,
2.5,
and
2.6
as
the
commensurator
of
the
closure
of
Π
v
[i.e.,
relative
to
the
interpretation
of
(a)]
inside
the
profinite
fundamental
group
of
†
D
±
relative
to
the
composite
poly-morphism
±
†
D
,v
(
†
φ
Θ
)
−1
v
−→
0
†
†
D
v
0
±
φ
Θ
v
ell
−→
0
†
D
±
†
Θ
determined
by
the
portions
of
†
φ
Θ
labeled
by
0
∈
T
,
v
∈
V
[cf.
±
,
φ
±
the
discussions
of
[IUTchI],
Examples
6.2,
(i);
6.3,
(i)],
and
ell
∼
±
cor
±
(c)
of
the
Δ
±
that
appears
in
Corollary
v
-outer
action
of
Δ
v
/
Δ
v
→
F
l
±
2.4,
(iii),
as
corresponding
to
the
F
l
-symmetry
of
[IUTchI],
Proposition
6.8,
(i).
Relative
to
the
interpretation
of
(a),
(b),
and
(c),
one
has
the
following
fundamental
observation
concerning
the
discussion
of
Remark
2.6.1:
the
single
basepoint
that
underlies
the
conjugate
synchronization
dis-
cussed
in
Remark
2.6.1
is
compatible
with
the
single
basepoint
that
underlies
the
label
synchronization
discussed
in
[IUTchI],
Remark
6.12.4.
That
is
to
say,
both
of
these
basepoints
may
be
thought
of
as
arising
from
a
single
basepoint
that
gives
rise
to
the
various
topological
groups
Π
v
,
Π
±
v
,
etc.
that
appear
in
Corollaries
2.4,
2.5,
and
2.6.
In
particular,
the
conjugate
synchronization
discussed
in
Remark
2.6.1
is
compat-
ible
with
the
F
±
l
-symmetry
of
[IUTchI],
Proposition
6.8,
(i)
[cf.
also
Remark
3.8.3
below].
Indeed,
this
compatibility
is
essentially
the
content
of
Corollary
2.4,
(iii)
[cf.
(c)
above].
(ii)
Note
that
the
compatibility
of
basepoints
discussed
in
(i)
contrasts
sharply
with
the
incompatibility
of
the
conjugate
synchronization
basepoint
of
Remark
2.6.1
with
the
F
l
-symmetry
of
[IUTchI],
Proposition
4.9,
(i),
in
the
case
of
D-ΘNF-Hodge
theaters.
At
a
more
concrete
level,
this
difference
between
F
±
l
-
and
F
l
-symmetries
may
be
understood
as
a
consequence
of
the
fact
that
whereas
the
F
±
l
-symmetry
is
defined
relative
to
a
single
copy
of
a
local
geometric
object
at
v
—
i.e.,
“
Π
±
v
”
[cf.
(a),
(b),
(c)
above]
—
the
F
l
-symmetry
involves
permuting
multiple
copies
of
local
geometric
objects
in
such
a
way
that
one
may
only
identify
these
multiple
copies
with
one
another
at
the
expense
of
allowing
the
phenomenon
of
“label
crushing”
[cf.
the
discussions
of
[IUTchI],
Remark
4.9.2,
(i),
(ii);
6.12.6,
(i),
(ii),
(iii)].
(iii)
Another
important
property
of
the
F
±
l
-symmetry
—
which
is
not
satisfied
±
by
the
F
l
-symmetry!
—
is
that
the
F
l
-symmetry
allows
comparison
with
the
label
zero
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.5],
hence,
in
particular,
comparison
with
the
copies
of
“O
×
”
[cf.
the
discussion
of
Remark
1.12.2]
that
k
occur
in
the
splittings
of
Corollary
1.12,
(ii),
that
give
rise
to
the
crucial
constant
multiple
rigidity
of
the
étale
theta
function.
This
important
property
is
precisely
the
content
of
Corollary
2.6.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
79
Remark
2.6.3.
(i)
The
discussion
of
independent
conjugacy
indeterminacies
in
Remark
2.5.2
and
of
“single
basepoints”
that
are
compatible
with
the
F
±
l
-symmetry
of
[IUTchI],
§6,
in
Remarks
2.6.1,
2.6.2
imply
rather
severe
restrictions
concerning
the
sub-
⊆
Γ
Ÿ
”
of
Remark
2.1.1,
(ii).
That
is
to
say,
suppose
that
one
attempts
graph
“Γ
Ÿ
to
develop
the
theory
of
the
present
§2
for
another
subgraph
Γ
of
the
graph
Γ
Ÿ
.
Recall
from
the
discussion
of
Remark
2.1.1,
(i),
that
the
graph
Γ
Ÿ
may
be
thought
of
as
a
“copy
of
the
real
line
R”,
in
which
the
integers
Z
⊆
R
are
taken
to
be
the
vertices,
and
the
line
segments
joining
the
integers
are
taken
to
be
the
edges.
Then
the
discussion
of
“single
basepoints”
[cf.
Remark
2.6.1]
implies,
first
of
all,
that
(a)
this
subgraph
Γ
must
be
connected.
Since,
moreover,
one
wishes
to
consider
the
crucial
splittings
of
Corollary
2.6,
(ii)
[cf.
Remark
2.6.2,
(iii)],
it
follows
that
(b)
this
subgraph
Γ
must
contain
the
vertex
of
Γ
Ÿ
labeled
“0”.
The
conditions
(a)
and
(b)
already
impose
substantial
restrictions
on
Γ
and
hence
on
the
collection
of
values
of
the
étale
theta
function
that
may
arise
by
restricting
to
the
μ
−
-translates
of
the
cusps
that
lie
in
Γ
[cf.
Remark
2.5.1,
(ii)]
—
i.e.,
on
the
collection
of
q
j
2
v
obtained
by
allowing
j
∈
Z
to
range
[relative
to
the
identification
of
the
vertices
of
Γ
Ÿ
with
the
integral
points
of
the
real
line]
over
the
“vertices”
of
Γ
[cf.
Remark
2.5.1,
(i)].
(ii)
By
abuse
of
notation,
let
us
write
“j
∈
Γ
”
for
“vertices”
j
∈
Z
that
lie
in
Γ
.
Also,
for
simplicity,
let
us
assume
that
the
subgraph
Γ
is
finite
[cf.
(iii)
below].
Then
ultimately,
in
the
theory
of
[IUTchIV],
when
we
consider
various
height
inequalities,
we
shall
be
concerned
with
the
issue
of
maximizing
the
quantity
def
min
{
j
2
}
||Γ
||
=
|Γ
|
−1
·
j∈F
l
j∈j
Γ
—
where
we
write
|Γ
|
for
the
cardinality
of
the
image
in
F
l
of
the
nonzero
elements
of
Γ
;
we
regard
the
“min”
over
an
empty
set
as
being
equal
to
zero;
we
think
of
the
various
j
∈
F
l
as
corresponding
to
the
subsets
of
Z
determined
by
the
fibers
of
the
natural
projection
Z
|F
l
|
(⊇
F
l
).
Here,
we
observe
that
(c)
the
set
of
“j’s”
that
occur
in
the
“min”
ranging
over
“j”
[i.e.,
not
over
“j”!]
that
appears
in
the
definition
of
||Γ
||
is
always
equal
to
a
fiber
of
the
restriction
to
the
set
of
vertices
of
Γ
of
the
natural
projection
Z
|F
l
|.
In
fact,
this
observation
(c)
is,
in
essence,
a
consequence
of
the
phenomenon
dis-
cussed
in
Remark
2.5.2
of
independent
conjugacy
indeterminacies
[cf.,
espe-
cially,
Remark
2.5.2,
(iv)]
—
i.e.,
roughly
speaking,
that
80
SHINICHI
MOCHIZUKI
one
cannot
restrict
the
étale
theta
function
to
“one
j
∈
Γ
”
without
also
restricting
the
étale
theta
function
to
the
various
“other
j
∈
Γ
”
that
lie
in
the
same
fiber
over
|F
l
|.
Next,
let
us
make
the
[easily
verified
—
cf.
(a),
(b)!]
observation
that
if
one
thinks
of
||Γ
||
as
a
function
of
|Γ
|,
then
as
|Γ
|
ranges
over
the
positive
integers,
it
holds
that
(d)
the
function
of
|Γ
|
constituted
by
||Γ
||
—
which
may
be
thought
of
as
a
sort
of
average
—
is
a
monotone
increasing
[but
not
strictly
increasing!]
function
of
|Γ
|
valued
in
the
positive
rational
numbers
which
attains
its
maximum
when
|Γ
|
=
l
and
is
constant
for
|Γ
|
≥
l
.
Now
it
follows
formally
from
(d)
that,
as
|Γ
|
ranges
over
the
positive
integers,
the
quantity
||Γ
||
attains
its
maximum
when
|Γ
|
=
l
—
hence,
in
particular,
when
.
Thus,
from
the
point
of
view
of
the
issue
of
maximizing
this
Γ
is
taken
to
be
Γ
Ÿ
[cf.
also
quantity
||Γ
||,
there
is
“no
loss
of
generality”
in
assuming
that
Γ
=
Γ
Ÿ
the
discussion
of
(iv)
below].
(iii)
Although
in
the
discussion
of
(ii)
above
we
assumed
that
Γ
is
finite,
this
assumption
does
not
in
fact
result
in
any
loss
of
generality.
Indeed,
one
verifies
immediately
that
||Γ
||
is
defined,
finite,
and
satisfies
the
evident
analogue
of
(d)
even
for
infinite
Γ
.
Thus,
the
case
of
infinite
Γ
may
be
excluded
without
loss
of
generality.
(iv)
Ultimately,
in
§4
of
the
present
paper,
we
shall
be
concerned
with
the
issue
of
globalizing,
via
the
construction
of
various
global
realified
Frobenioids,
the
monoids
determined
by
the
theta
values
at
v
∈
V
bad
that
appear
in
the
present
§2.
This
globalization
will
be
achieved,
in
effect,
by
imposing
the
condition
that
the
product
formula
be
satisfied.
On
the
other
hand,
the
indeterminacies
discussed
in
(ii)
above
[cf.,
especially,
(ii),
(c)]
that
arise
when
a
fiber
of
Γ
over
|F
l
|
contains
more
than
one
element
are
easily
seen
to
be
fundamentally
incompatible
with
the
product
formula.
In
particular,
from
the
point
of
view
of
the
issue
of
maximiz-
ing
the
quantity
||Γ
||,
in
fact
the
only
choice
for
Γ
that
is
compatible
with
the
.
“globalization
via
the
product
formula”
to
be
performed
in
§4
is
Γ
Ÿ
(v)
One
may
summarize
the
discussion
of
(i),
(ii),
(iii),
and
(iv)
as
follows:
j
2
the
collection
of
values
“q
”
of
the
étale
theta
function
determined
by
the
v
subgraph
Γ
Ÿ
is
of
a
highly
distinguished
nature
—
and,
indeed,
is
essentially
determined
[cf.
the
discussion
at
the
end
of
(ii);
the
discussion
of
(iv)]
by
the
requirement
of
maximizing
the
quantity
“||Γ
||”
in
a
fashion
compatible
with
the
global
product
formula,
together
with
various
qualitative
considerations
that
arise
from
Corollaries
2.4,
2.5,
2.6;
the
discussion
of
Remarks
2.5.1,
2.5.2,
2.6.1,
2.6.2.
Definition
2.7.
In
the
notation
of
Definition
2.3:
Let
M
Θ
∗
=
Θ
{.
.
.
→
M
Θ
M
→
M
M
→
.
.
.
}
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
81
be
a
projective
system
of
mono-theta
environments
as
in
Proposition
1.5,
such
that
∼
Π
X
(M
Θ
∗
)
=
Π
v
.
(i)
Write
Π
M
Θ
∗
for
the
inverse
limit
of
the
induced
projective
system
of
topological
groups
{.
.
.
→
→
.
.
.
}
[cf.
the
notation
discussed
at
the
beginning
of
Definition
Π
M
Θ
→
Π
M
Θ
M
M
1.1].
Thus,
[in
the
notation
of
Proposition
1.5]
we
have
a
natural
homomorphism
of
topological
groups
→
Π
X
(M
Θ
Π
M
Θ
∗
)
∗
whose
kernel
may
be
identified
with
the
exterior
cyclotome
Π
μ
(M
Θ
∗
),
and
whose
tp
Θ
∼
image
is
the
subgroup
of
Π
X
(M
∗
)
=
Π
v
determined
by
Π
Y
.
v
(ii)
Write
⊆
Π
M
Θ
Π
M
Θ
¨
⊆
Π
M
Θ
∗
∗
∗
Θ
∼
for
the
respective
inverse
images
of
Π
v
;
¨
⊆
Π
v
⊆
Π
v
=
Π
X
(M
∗
)
in
Π
M
Θ
∗
(l
·
Δ
Θ
)(M
Θ
¨
),
∗
Π
μ
(M
Θ
¨
),
∗
Θ
Π
v
¨
(M
∗
¨
),
G
v
(M
Θ
¨
)
∗
for
the
subquotients
of
Π
M
Θ
¨
determined
by
the
subquotient
Π
μ
(M
Θ
and
∗
)
of
Π
M
Θ
∗
∗
Θ
the
subquotients
(l
·
Δ
Θ
)(Π
X
(M
Θ
¨
,
and
G
v
(Π
X
(M
∗
))
∗
))
[cf.
Proposition
1.4],
Π
v
[cf.
Corollary
2.5,
(i)]
of
Π
v
∼
=
Π
X
(M
Θ
∗
).
Thus,
we
obtain
a
cyclotomic
rigidity
isomorphism
∼
Θ
(l
·
Δ
Θ
)(M
Θ
¨
)
→
Π
μ
(M
∗
¨
)
∗
∼
Θ
—
i.e.,
by
restricting
the
cyclotomic
rigidity
isomorphism
(l
·
Δ
Θ
)(M
Θ
∗
)
→
Π
μ
(M
∗
)
of
Proposition
1.5,
(iii),
to
Π
M
Θ
¨
.
∗
Corollary
2.8.
(Mono-theta-theoretic
Theta
Evaluation)
In
the
notation
of
Definition
2.7:
Suppose
that
we
are
in
the
situation
of
Proposition
2.2,
(ii);
Corollary
2.5,
(ii);
to
simplify
notation,
we
assume
that
Π
X
(M
Θ
∗
)
=
Π
v
,
and
we
use
the
notation
for
objects
constructed
from
“Π
v
”
to
denote
the
corresponding
objects
constructed
from
Π
X
(M
Θ
∗
).
Also,
let
us
write
γ
(M
Θ
∗
)
for
the
projective
system
of
mono-theta
environments
obtained
via
transport
of
∼
structure
from
the
isomorphism
Π
v
→
Π
γv
determined
by
conjugation
by
γ.
(i)
(Restriction
of
Étale
Theta
Functions
to
Subgraphs
and
Evalu-
ation
Points)
In
the
situation
of
Proposition
2.2,
(ii);
Corollary
2.5,
(ii),
let
us
apply
the
cyclotomic
rigidity
isomorphisms
∼
γ
Θ
γ
(l
·
Δ
Θ
)((M
Θ
¨
)
)
→
Π
μ
((M
∗
¨
)
);
∗
∼
γ
Θ
γ
(l
·
Δ
Θ
)((M
Θ
∗
)
)
→
Π
μ
((M
∗
)
)
82
SHINICHI
MOCHIZUKI
γ
[cf.
Definition
2.7,
(ii),
applied
to
(M
Θ
∗
)
]
to
replace
“(l
·
Δ
Θ
)(−)”
by
“Π
μ
(−)”.
ι
Then
the
ι
γ
-invariant
subsets
θ
(Π
γv
)
⊆
θ(Π
γv
),
∞
θ
ι
(Π
γv
)
⊆
∞
θ(Π
γv
)
[cf.
Proposition
2.2,
(ii);
Corollary
2.5,
(ii)]
determine
ι
γ
-invariant
subsets
γ
θ
ι
env
((M
Θ
∗
)
)
⊆
γ
θ
env
((M
Θ
∗
)
);
ι
Θ
γ
∞
θ
env
((M
∗
)
)
⊆
Θ
γ
∞
θ
env
((M
∗
)
)
ι
γ
Θ
γ
[cf.
Proposition
1.5,
(iii),
applied
to
(M
Θ
∗
)
];
restriction
of
these
subsets
θ
env
((M
∗
)
),
ι
Θ
γ
Θ
γ
¨
((M
∗
∞
θ
env
((M
∗
)
)
to
Π
v
¨
)
)
yields
μ
2l
-,
μ-orbits
of
elements
γ
θ
ι
env
((M
Θ
¨
)
)
∗
⊆
ι
Θ
γ
∞
θ
env
((M
∗
¨
)
)
⊆
1
Θ
γ
Θ
γ
lim
¨
((M
∗
¨
)
)|
J
,
Π
μ
((M
∗
¨
)
))
−→
H
(Π
v
J
—
where
J
⊆
Π
v
ranges
over
the
open
subgroups
of
Π
v
—
which,
upon
further
δ
of
Corollary
2.4,
(ii),
(c),
yield
restriction
to
the
decomposition
groups
D
t,μ
−
μ
2l
-,
μ-orbits
of
elements
γ
θ
t
env
((M
Θ
¨
)
)
∗
⊆
t
Θ
γ
∞
θ
env
((M
∗
¨
)
)
⊆
1
Θ
γ
Θ
γ
lim
¨
)
)|
J
G
,
Π
μ
((M
∗
¨
)
))
−→
H
(G
v
((M
∗
J
G
∼
γ
for
each
t
∈
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
—
where
J
G
⊆
G
v
((M
Θ
¨
)
)
ranges
∗
∼
γ
over
the
open
subgroups
of
G
v
((M
Θ
¨
)
);
the
“
→
”
is
induced
by
conjugation
by
γ.
∗
t
γ
Θ
γ
Moreover,
the
sets
θ
t
env
((M
Θ
¨
)
),
∞
θ
env
((M
∗
¨
)
)
depend
only
on
the
label
|t|
∈
|F
l
|
∗
γ
determined
by
t
[cf.
Corollary
2.5,
(ii)].
Thus,
we
shall
write
θ
|t|
((M
Θ
¨
)
)
=
∗
env
def
|t|
t
γ
Θ
γ
Θ
γ
θ
t
env
((M
Θ
¨
)
),
∞
θ
env
((M
∗
¨
)
)
=
∞
θ
env
((M
∗
¨
)
).
∗
def
(ii)
(Functorial
Group-theoretic
Evaluation
Algorithm)
If
one
starts
Θ
γ
Θ
with
an
arbitrary
Δ
±
¨
((M
∗
¨
(M
∗
v
-conjugate
Π
v
¨
)
)
of
Π
v
¨
),
and
one
consid-
∼
±
ers,
as
t
ranges
over
the
elements
of
LabCusp
(Π
γv
)
→
LabCusp
±
(Π
v
)
[where
the
∼
γ
“
→
”
is
induced
by
conjugation
by
γ],
the
resulting
μ
2l
-,
μ-orbits
θ
|t|
((M
Θ
¨
)
),
∗
env
|t|
Θ
γ
±
δ
∞
θ
env
((M
∗
¨
)
)
arising
from
an
arbitrary
Δ
v
-conjugate
I
t
of
I
t
that
is
con-
Θ
γ
tained
in
Π
v
¨
((M
∗
¨
)
)
[cf.
(i)],
then
one
obtains
an
algorithm
for
constructing
the
collections
of
μ
2l
-,
μ-orbits
γ
{θ
|t|
((M
Θ
¨
)
)}
|t|∈|F
l
|
;
∗
env
γ
{
∞
θ
|t|
((M
Θ
¨
)
)}
|t|∈|F
l
|
∗
env
which
is
functorial
in
the
projective
system
of
mono-theta
environments
M
Θ
∗
and,
±
moreover,
compatible
with
the
independent
conjugacy
actions
of
Δ
v
on
the
Θ
γ
2
Θ
γ
2
sets
{I
t
γ
1
}
γ
∈
Π
±
=
{I
t
γ
1
}
γ
∈
Δ
±
and
{Π
v
¨
((M
∗
¨
((M
∗
¨
)
)}
γ
∈
Π
±
=
{Π
v
¨
)
)}
γ
∈
Δ
±
1
v
1
v
2
v
2
v
[cf.
the
sets
of
Corollary
2.4,
(i);
Remark
2.2.1].
(iii)
(Splittings
at
Zero-labeled
Evaluation
Points)
In
the
situation
of
(i),
suppose
that
t
is
taken
to
be
the
zero
element.
Then,
by
applying
the
cy-
clotomic
rigidity
isomorphisms
of
(i)
to
replace
“(l
·
Δ
Θ
)(−)”
by
“Π
μ
(−)”
—
an
??
operation
that,
when
applied
to
“M
TM
(−)”
[where
“??”
∈
{×,
μ,
×μ}],
we
shall
γ
Θ
γ
denote
by
replacing
the
notation
“Π
v
¨
)
”
—
in
Corollary
2.6,
(ii),
the
¨
”
by
“(M
∗
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
83
second
restriction
operation
discussed
in
(i)
determines
splittings
[cf.
Corollary
2.6,
(ii)]
×μ
μ
ι
γ
Θ
γ
Θ
γ
((M
Θ
M
TM
¨
)
)
×
{
∞
θ
env
((M
∗
¨
)
)/M
TM
((M
∗
¨
)
)}
∗
μ
×
γ
Θ
γ
·
∞
θ
ι
env
((M
Θ
of
M
TM
¨
)
)/M
TM
((M
∗
¨
)
)
which
are
compatible,
relative
to
the
first
∗
restriction
operation
discussed
in
(i),
with
the
splittings
of
Corollary
1.12,
(ii)
[i.e.,
∼
Θ
relative
to
any
isomorphism
M
Θ
∗
→
M
∗
(Π
v
)
—
cf.
Proposition
1.2,
(i);
Proposition
1.5,
(i);
Remarks
2.8.1,
2.8.2
below].
Proof.
Assertions
(i),
(ii),
and
(iii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.8.1.
One
may
regard
Corollaries
2.5,
2.6
as
a
special
case
of
Corollary
2.8,
i.e.,
the
case
where
the
projective
system
of
mono-theta
environments
M
Θ
∗
arises
from
the
topological
group
Π
v
by
applying
the
functorial
group-theoretic
algorithm
of
Proposition
1.2,
(i)
[cf.
also
Proposition
1.5,
(i)].
Remark
2.8.2.
The
significance
of
the
mono-theta-theoretic
version
of
Corol-
laries
2.5,
2.6
constituted
by
Corollary
2.8
lies
in
the
fact
that
this
mono-theta-
theoretic
version
allows
one
to
relate
the
group-theoretic
theta
evaluation
theory
of
the
present
§2
to
the
theory
of
Frobenioid-theoretic
theta
functions
associ-
ated
to
tempered
Frobenioids
[cf.
[EtTh],
§5],
i.e.,
by
considering
the
case
where
M
Θ
∗
arises
from
a
tempered
Frobenioid
[cf.
Proposition
1.2,
(ii)].
For
instance,
by
considering
the
case
where
M
Θ
∗
arises
from
a
tempered
Frobenioid,
one
may
treat
the
Frobenioid-theoretic
cyclotomes
[i.e.,
cyclotomes
that
arise
from
the
units
of
the
Frobenioid]
of
Proposition
1.3,
(i),
in
the
context
of
the
theory
of
the
present
§2.
Remark
2.8.3.
(i)
The
use
of
the
archimedean
line
segment
Γ
X
⊆
Γ
X
[cf.
Remark
2.1.1,
(ii)]
to
single
out
the
elements
∈
{−l
,
−l
+
1,
.
.
.
,
−1,
0,
1,
.
.
.
,
l
−
1,
l
}
—
i.e.,
the
elements
with
absolute
value
≤
l
—
within
the
nonarchimedean
congruence
classes
modulo
l
constituted
by
an
element
∈
F
l
is
reminiscent
of
the
computation
of
the
set
of
global
sections
of
an
arithmetic
line
bundle
on
a
number
field
[cf.,
e.g.,
[Szp],
pp.
13-14],
as
well
as
of
the
arithmetic
inherent
in
the
graph
theory
associated
to
the
loop
Γ
X
[cf.
[SemiAnbd],
Remark
1.5.1].
(ii)
The
sort
of
argument
discussed
in
(i)
involving
the
connected,
“archime-
dean”
line
segment
Γ
X
⊆
Γ
X
[cf.
Remark
2.6.1
for
more
on
the
importance
of
this
connectedness]
depends,
in
an
essential
way,
on
the
discreteness
of
Z
(
∼
=
Z).
Put
another
way,
this
sort
of
argument
may
be
thought
of
as
an
application
of
the
discrete
rigidity
that
forms
one
of
the
central
themes
of
[EtTh].
Note,
moreover,
that
in
the
context
of
Corollary
2.8,
this
application
of
discrete
rigidity
is
closely
related
to
the
application
of
cyclotomic
rigidity.
This
is
perhaps
not
so
surpris-
ing,
since
discrete
rigidity
—
in
the
form
of
the
discreteness
of
squares
of
elements
of
Z,
i.e.,
in
effect,
the
quotient
of
Z
by
the
action
of
{±1}
—
may
be
thought
of
as
a
sort
of
dual
property
to
the
cyclotomic
rigidity
of
“(l
·
Δ
Θ
)(−)”.
Indeed,
one
84
SHINICHI
MOCHIZUKI
may
think
of
this
duality
as
being
embodied
in
the
very
structure
and
values
of
the
étale
theta
function
[cf.
[EtTh],
Proposition
1.4,
(ii),
(iii);
[EtTh],
Proposition
1.5,
(ii)].
In
a
similar
vein,
one
may
also
consider
the
theory
of
group-theoretic
theta
evaluation
developed
in
the
present
§2
in
the
context
of
the
natural
isomorphism
∼
“μ
Z
(G
k
)
→
μ
Z
(Π
X
)”
of
[AbsTopIII],
Corollary
1.10,
(c)
[cf.
also
Proposition
1.3,
(ii);
Corollary
1.11,
(b)].
Corollary
2.9.
(Theta
Evaluation
via
Base-field-theoretic
Cyclotomes)
Suppose
that
we
are
in
the
situation
of
Proposition
2.2,
(ii);
Corollary
2.5,
(ii).
Also,
let
us
write
∼
∼
γ
μ
Z
(G
v
(Π
γv
¨
))
→
(l
·
Δ
Θ
)(Π
v
¨
)
μ
Z
(G
v
(Π
v
))
→
(l
·
Δ
Θ
)(Π
v
);
for
the
cyclotomic
rigidity
isomorphisms
determined
by
the
natural
isomor-
∼
phism
“μ
Z
(G
k
)
→
μ
Z
(Π
X
)”
of
[AbsTopIII],
Corollary
1.10,
(c)
[cf.
also
Propo-
sition
1.3,
(ii);
Corollary
1.11,
(b)]
and
its
restriction
to
Π
γv
¨
[cf.
Corollary
2.5,
(i)].
(i)
(Restriction
of
Étale
Theta
Functions
to
Subgraphs
and
Eval-
uation
Points)
In
the
situation
of
Proposition
2.2,
(ii);
Corollary
2.5,
(ii),
let
us
apply
the
above
cyclotomic
rigidity
isomorphisms
to
replace
“(l
·
Δ
Θ
)(−)”
by
“μ
Z
(G
v
(−))”.
Then
the
ι
γ
-invariant
subsets
θ
ι
(Π
γv
)
⊆
θ(Π
γv
),
∞
θ
ι
(Π
γv
)
⊆
∞
θ(Π
γv
)
[cf.
Proposition
2.2,
(ii);
Corollary
2.5,
(ii)]
determine
ι
γ
-invariant
subsets
θ
ι
bs
(Π
γv
)
θ
bs
(Π
γv
);
⊆
ι
γ
∞
θ
bs
(Π
v
)
⊆
γ
∞
θ
bs
(Π
v
)
—
where
one
may
think
of
the
“bs”
as
an
abbreviation
of
the
term
“base-field-
theoretic”;
restriction
of
these
subsets
θ
ι
bs
(Π
γv
),
∞
θ
ι
bs
(Π
γv
)
to
Π
γv
¨
yields
μ
2l
-,
μ-orbits
of
elements
θ
ι
bs
(Π
γv
¨
)
⊆
γ
ι
∞
θ
bs
(Π
v
¨
)
γ
γ
1
lim
¨
|
J
,
μ
Z
(G
v
(Π
v
¨
)))
−→
H
(Π
v
⊆
J
—
where
J
⊆
Π
v
ranges
over
the
open
subgroups
of
Π
v
—
which,
upon
further
δ
of
Corollary
2.4,
(ii),
(c),
yield
restriction
to
the
decomposition
groups
D
t,μ
−
μ
2l
-,
μ-orbits
of
elements
θ
t
bs
(Π
γv
¨
)
⊆
γ
t
∞
θ
bs
(Π
v
¨
)
⊆
γ
γ
1
lim
¨
)|
J
G
,
μ
Z
(G
v
(Π
v
¨
)))
−→
H
(G
v
(Π
v
J
G
∼
for
each
t
∈
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
—
where
J
G
⊆
G
v
(Π
γv
¨
)
ranges
over
∼
the
open
subgroups
of
G
v
(Π
γv
¨
);
the
“
→
”
is
induced
by
conjugation
by
γ.
Moreover,
γ
t
the
sets
θ
t
bs
(Π
γv
¨
),
∞
θ
bs
(Π
v
¨
)
depend
only
on
the
label
|t|
∈
|F
l
|
determined
by
t
γ
γ
t
|t|
(Π
γv
[cf.
Corollary
2.5,
(ii)].
Thus,
we
shall
write
θ
|t|
¨
)
=
θ
bs
(Π
v
¨
),
∞
θ
bs
(Π
v
¨
)
=
bs
def
γ
t
∞
θ
bs
(Π
v
¨
).
def
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
85
(ii)
(Functorial
Group-theoretic
Evaluation
Algorithm)
If
one
starts
γ
with
an
arbitrary
Δ
±
¨
,
and
one
considers,
as
t
ranges
v
-conjugate
Π
v
¨
of
Π
v
∼
∼
over
the
elements
of
LabCusp
±
(Π
γv
)
→
LabCusp
±
(Π
v
)
[where
the
“
→
”
is
induced
γ
|t|
by
conjugation
by
γ],
the
resulting
μ
2l
-,
μ-orbits
θ
|t|
(Π
γv
¨
),
∞
θ
bs
(Π
v
¨
)
arising
from
bs
γ
δ
an
arbitrary
Δ
±
v
-conjugate
I
t
of
I
t
that
is
contained
in
Π
v
¨
[cf.
(i)],
then
one
obtains
an
algorithm
for
constructing
the
collections
of
μ
2l
-,
μ-orbits
{θ
|t|
(Π
γv
¨
)}
|t|∈|F
l
|
;
bs
{
∞
θ
|t|
(Π
γv
¨
)}
|t|∈|F
l
|
bs
which
is
functorial
in
the
topological
group
Π
v
and,
moreover,
compatible
with
γ
1
γ
1
the
independent
conjugacy
actions
of
Δ
±
v
on
the
sets
{I
t
}
γ
∈
Π
±
=
{I
t
}
γ
∈
Δ
±
1
v
1
v
γ
2
and
{Π
γ
v
2
¨
}
γ
∈
Π
±
=
{Π
v
¨
}
γ
∈
Δ
±
[cf.
the
sets
of
Corollary
2.4,
(i);
Remark
2.2.1].
2
v
2
v
(iii)
(Splittings
at
Zero-labeled
Evaluation
Points)
In
the
situation
of
(i),
suppose
that
t
is
taken
to
be
the
zero
element.
Then,
by
applying
the
cyclo-
tomic
rigidity
isomorphisms
reviewed
above
to
replace
“(l·Δ
Θ
)(−)”
by
“μ
Z
(G
v
(−))”
??
(−)”
[where
“??”
∈
{×,
μ,
×μ}],
we
—
an
operation
that,
when
applied
to
“M
TM
shall
denote
by
means
of
a
subscript
“bs”
—
in
Corollary
2.6,
(ii),
the
second
restriction
operation
discussed
in
(i)
determines
splittings
[cf.
Corollary
2.6,
(ii)]
×μ
γ
μ
γ
ι
(Π
γv
M
TM
¨
)
bs
×
{
∞
θ
bs
(Π
v
¨
)/M
TM
(Π
v
¨
)
bs
}
μ
γ
×
·
∞
θ
ι
bs
(Π
γv
of
M
TM
¨
)/M
TM
(Π
v
¨
)
bs
which
are
compatible,
relative
to
the
first
re-
striction
operation
discussed
in
(i)
and
the
cyclotomic
rigidity
isomorphisms
re-
viewed
above,
with
the
splittings
of
Corollary
1.12,
(ii).
Proof.
Assertions
(i),
(ii),
and
(iii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
2.9.1.
(i)
Let
us
recall
that
[the
cyclotomic
rigidity
isomorphisms
involving]
the
cyclo-
tomes
“Π
μ
(−)”
that
appear
in
Corollary
2.8
admit
a
multiradial
formulation
[cf.
Corollary
1.10].
By
contrast,
at
least
relative
to
the
point
of
view
of
Remark
1.11.3,
(iv),
[the
cyclotomic
rigidity
isomorphisms
involving]
the
cyclotomes
“μ
Z
(G
v
(−))”
that
appear
in
Corollary
2.9
only
admit
a
uniradial
formulation
—
i.e.,
unless
one
is
willing
to
sacrifice
the
crucial
cyclotomic
rigidity
under
consideration
as
in
the
formulation
of
Corollary
1.11.
(ii)
On
the
other
hand,
the
use
of
[the
cyclotomic
rigidity
isomorphisms
involv-
ing]
the
cyclotomes
“μ
Z
(G
v
(−))”
has
the
crucial
advantage
that
it
allows
one
to
apply
the
[not
multiradially
(!),
but
rather]
uniradially
defined
natural
surjection
H
1
(G
v
(−),
μ
Z
(G
v
(−)))
Z
of
Remark
1.11.5,
(i),
(ii).
86
SHINICHI
MOCHIZUKI
(iii)
One
immediate
consequence
of
the
discussion
of
(i)
is
the
observation
that,
at
least
relative
to
the
point
of
view
of
Remark
1.11.3,
(iv),
the
algorithms
of
Corollary
2.9,
(ii),
(iii),
only
give
rise
to
a
uniradially
defined
functor.
On
the
other
hand,
one
important
consequence
of
the
theory
to
be
developed
in
[IUTchIII]
is
the
result
that,
by
applying
the
theory
of
log-shells
[cf.
[AbsTopIII]],
one
may
modify
these
algorithms
in
such
a
way
as
to
obtain
algorithms
that
[yield
functors
which]
are
manifestly
multiradially
defined
—
albeit
at
the
cost
of
allowing
for
certain
[relatively
mild!]
indeterminacies.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
87
Section
3:
Tempered
Gaussian
Frobenioids
In
the
present
§3,
we
relate
the
theory
of
group-theoretic
algorithms
surround-
ing
the
Hodge-Arakelov-theoretic
evaluation
of
the
étale
theta
function
on
l-torsion
points
developed
in
§1,
§2
to
the
local
portion
at
bad
primes
[i.e.,
at
v
∈
V
bad
]
of
the
various
Frobenioids
considered
in
[IUTchI],
§3,
§4,
§5,
§6.
In
par-
ticular,
we
shall
discuss
how
the
various
multiradial
formulations
developed
in
§1
and
the
theory
of
conjugate
synchronization
developed
in
§2
may
be
applied
in
the
context
of
the
“tempered
Gaussian
Frobenioids”
that
arise
from
the
Hodge-Arakelov-theoretic
evaluation
of
the
étale
theta
function
on
l-torsion
points.
In
the
present
§3,
we
shall
continue
to
use
the
notation
of
§2.
In
particular,
our
discussion
concerns
the
local
portion
at
v
∈
V
bad
of
the
various
mathematical
objects
considered
in
[IUTchI],
§3,
§4,
§5,
§6.
Proposition
3.1.
(Mono-theta-theoretic
Theta
Monoids)
Let
M
Θ
∗
=
Θ
{.
.
.
→
M
Θ
M
→
M
M
→
.
.
.
}
be
a
projective
system
of
mono-theta
environments
[cf.
Proposition
1.5,
∼
Corollary
2.8]
such
that
Π
X
(M
Θ
∗
)
=
Π
v
.
In
the
following,
to
simplify
the
notation,
we
shall
denote
a
“Π
X
(M
Θ
∗
)”
in
parenthesis
by
means
of
the
abbreviated
notation
Θ
“M
∗
”.
(i)
(Split
Theta
Monoids)
By
applying
the
constructions
of
Proposition
1.5,
(iii);
Corollary
2.8,
(i)
[cf.
also
Corollary
1.12,
(d)],
one
obtains
a
functorial
algorithm
ι
ι
×
Θ
Θ
→
M
TM
(M
Θ
M
Θ
∗
∗
),
θ
env
(M
∗
),
∞
θ
env
(M
∗
),
ι
×
Θ
1
Θ
Θ
M
TM
·
∞
θ
env
(M
∗
)
⊆
lim
−→
H
(Π
Ÿ
(M
∗
)|
J
,
Π
μ
(M
∗
))
ι
J
—
where
J
ranges
over
the
finite
index
open
subgroups
of
Π
X
(M
Θ
∗
),
and
ι
ranges
over
the
inversion
automorphisms
of
Proposition
2.2,
(i)
—
for
constructing
var-
ious
subsets
of
the
direct
limit
of
cohomology
modules
in
the
above
display;
this
collection
of
subsets
is
equipped
with
a
natural
conjugation
action
by
Π
X
(M
Θ
∗
).
In
particular,
one
obtains
a
functorial
algorithm
for
constructing
the
data
def
ι
×
Θ
ι
Θ
Θ
Θ
N
Ψ
env
(M
∗
)
=
Ψ
env
(M
∗
)
=
M
TM
(M
∗
)
·
θ
env
(M
∗
)
;
ι
def
ι
×
Θ
ι
Θ
Θ
N
Θ
∞
Ψ
env
(M
∗
)
=
∞
Ψ
env
(M
∗
)
=
M
TM
(M
∗
)
·
∞
θ
env
(M
∗
)
ι
ι
Θ
consisting
of
the
submonoids
{Ψ
ι
env
(M
Θ
∗
)}
ι
,
{
∞
Ψ
env
(M
∗
)}
ι
[of
the
direct
limit
of
cohomology
modules
in
the
first
display
of
the
present
(i)]
generated,
respectively,
ι
×
×
Θ
by
the
subsets
“M
TM
·
θ
ι
env
(M
Θ
∗
)”,
“M
TM
·
∞
θ
env
(M
∗
)”,
as
well
as
a
functorial
algorithm
for
constructing
the
splittings
up
to
torsion
determined
by
the
subsets
ι
ι
×
Θ
Θ
“M
TM
(M
Θ
∗
)”,
“θ
env
(M
∗
)”,
“
∞
θ
env
(M
∗
)”
[cf.
Corollary
2.8,
(iii)].
We
shall
refer
ι
Θ
to
each
Ψ
ι
env
(M
Θ
∗
),
∞
Ψ
env
(M
∗
)
as
a
theta
monoid.
88
SHINICHI
MOCHIZUKI
(ii)
(Constant
Monoids)
By
applying
the
cyclotomic
rigidity
isomor-
phisms
of
Corollaries
2.8,
(i);
2.9,
and
considering
the
inverse
image
of
Z
⊆
Z
Θ
via
the
surjection
of
Remark
1.11.5,
(i),
applied
to
G
v
(M
Θ
∗
)
(=
G
v
(Π
X
(M
∗
)))
[cf.
the
notation
of
Corollary
2.5,
(i)],
one
obtains
a
functorial
algorithm
M
Θ
∗
def
Θ
1
Θ
Θ
→
Ψ
cns
(M
Θ
∗
)
=
M
TM
(M
∗
)
⊆
lim
−→
H
(Π
Ÿ
(M
∗
)|
J
,
Π
μ
(M
∗
))
J
[where
J
is
as
in
(i)]
for
constructing
a
“monoid
of
constants”
—
i.e.,
which
is
naturally
isomorphic
to
O
F
[cf.
Example
1.8,
(ii)]
—
equipped
with
a
natural
v
Θ
conjugation
action
by
Π
X
(M
Θ
∗
).
We
shall
refer
to
Ψ
cns
(M
∗
)
as
a
constant
monoid.
Proof.
Assertions
(i)
and
(ii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Before
proceeding,
we
pause
to
review
the
theory
of
tempered
Frobenioids
dis-
cussed
in
[IUTchI],
Example
3.2.
Example
3.2.
Theta
Monoids
Constructed
from
Tempered
Frobenioids.
In
the
situation
of
[IUTchI],
Example
3.2:
(i)
Recall
the
tempered
Frobenioid
F
v
of
[IUTchI],
Example
3.2,
(i),
(ii),
(v)
[cf.
also
[IUTchI],
Remark
3.2.3,
(i),
(ii)].
Then,
in
the
notation
of
loc.
cit.,
the
choice
of
a
Frobenioid-theoretic
theta
function
Θ
v
∈
O
×
(T
÷
)
Ÿ
v
—
i.e.,
among
the
μ
2l
(T
÷
)-multiples
of
the
Aut
D
v
(
Ÿ
v
)-conjugates
of
Θ
v
—
deter-
Ÿ
v
mines
a
monoid
O
C
Θ
(−)
on
D
v
Θ
.
Now
suppose,
for
simplicity,
that
the
topological
v
group
Π
v
arises
from
a
basepoint,
i.e.,
more
concretely,
from
a
“universal
covering
pro-object”
A
Θ
∞
of
D
v
[i.e.,
a
pro-object
determined
by
a
cofinal
projective
system
of
Galois
objects
of
D
v
].
Then
by
evaluating
O
C
Θ
(−)
on
[the
“universal
covering
v
pro-object”
of
D
v
Θ
determined
by]
A
Θ
∞
,
we
obtain
submonoids
[in
the
usual
sense]
Ψ
F
v
Θ
,id
=
O
C
Θ
(A
Θ
∞
)
⊆
∞
Ψ
F
v
Θ
,id
def
v
=
def
=
N
O
C
×
Θ
(A
Θ
∞
)
·
Θ
v
|
A
Θ
∞
v
Q
≥0
O
C
×
Θ
(A
Θ
|
A
Θ
∞
)
·
Θ
v
∞
v
⊆
O
×
(T
÷
)
A
Θ
∞
—
where
the
superscript
“Q
≥0
”
denotes
the
set
of
elements
for
which
some
[positive
.
In
a
similar
vein,
integer]
power
is
equal
to
a
[positive
integer]
power
of
Θ
v
|
A
Θ
∞
by
considering
[cf.
[IUTchI],
Remark
3.2.3,
(i)]
the
various
conjugates
Θ
α
of
Θ
v
,
v
for
α
∈
Aut
D
v
(
Ÿ
v
),
we
also
obtain
submonoids
Ψ
F
v
Θ
,α
⊆
∞
Ψ
F
v
Θ
,α
⊆
O
×
(T
÷
).
A
Θ
∞
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
89
Moreover,
one
has
a
natural
surjection
Π
v
Aut
D
v
(
Ÿ
v
),
as
well
as
a
natural
conjugation
action
of
Π
v
on
the
collections
of
submonoids
def
def
=
Ψ
F
v
Θ
,α
;
∞
Ψ
F
v
Θ
=
Ψ
F
v
Θ
∞
Ψ
F
v
Θ
,α
α∈Π
v
α∈Π
v
—
i.e.,
where,
by
abuse
of
notation,
we
think
of
the
subscripted
“α’s”
as
denoting
the
image
of
“α”
via
the
surjection
Π
v
Aut
D
v
(
Ÿ
v
).
Also,
we
recall
from
loc.
cit.
≥0
that
Θ
Q
|
A
Θ
determines
characteristic
splittings,
up
to
torsion,
of
the
monoids
∞
v
Ψ
F
v
Θ
,α
[cf.
the
“τ
v
Θ
”
of
[IUTchI],
Example
3.2,
(v)],
∞
Ψ
F
v
Θ
,α
which
are
compatible
with
the
action
of
Π
v
.
Finally,
we
recall
that
the
collection
of
data
,
∞
Ψ
F
v
Θ
=
∞
Ψ
F
v
Θ
,α
Π
v
Ψ
F
v
Θ
=
Ψ
F
v
Θ
,α
α∈Π
v
α∈Π
v
—
i.e.,
consisting
of
two
collections
of
submonoids
of
the
group
of
units
[namely,
O
×
(T
÷
)]
associated
to
the
birationalization
of
a
certain
characteristic
pro-object
A
Θ
∞
of
F
v
,
equipped
with
the
conjugation
action
by
an
automorphism
group
of
a
certain
characteristic
pro-object
of
D
v
—
as
well
as
the
characteristic
splittings,
up
to
torsion,
just
discussed,
may
be
reconstructed
category-theoretically
from
F
v
[cf.
[IUTchI],
Example
3.2,
(vi),
(e)],
up
to
an
indeterminacy
arising
from
the
inner
automorphisms
of
Π
v
.
(ii)
In
a
similar,
but
somewhat
simpler,
vein,
the
Frobenioid
structure
on
the
subcategory
C
v
⊆
F
v
—
i.e.,
the
“base-field-theoretic
hull”
of
the
tempered
Frobenioid
F
v
[cf.
[IUTchI],
Example
3.2,
(iii)]
—
determines,
via
the
general
theory
of
Frobenioids
[cf.
[FrdI],
Proposition
2.2],
a
monoid
O
C
v
(−)
on
D
v
.
Then
by
evaluating
O
C
v
(−)
on
A
Θ
∞
,
we
obtain
a
monoid
[in
the
usual
sense]
Ψ
C
v
def
=
O
C
v
(A
Θ
∞
)
which
is
equipped
with
a
natural
action
by
Π
v
.
Finally,
we
recall
that
the
collection
of
data
Π
v
Ψ
C
v
—
i.e.,
consisting
of
a
submonoid
of
the
group
of
units
[namely,
O
×
(T
÷
)]
associ-
A
Θ
∞
ated
to
the
birationalization
of
a
certain
characteristic
pro-object
of
F
v
,
equipped
with
the
conjugation
action
by
an
automorphism
group
of
a
certain
characteristic
pro-object
of
D
v
—
may
be
reconstructed
category-theoretically
from
F
v
[cf.
[IUTchI],
Example
3.2,
(iii);
[IUTchI],
Example
3.2,
(vi),
(d);
[FrdI],
Theorem
3.4,
(iv);
[FrdII],
Theorem
1.2,
(i);
[FrdII],
Example
1.3,
(i)],
up
to
an
indeterminacy
arising
from
the
inner
automorphisms
of
Π
v
.
Proposition
3.3.
(Frobenioid-theoretic
Theta
Monoids)
Suppose,
in
the
situation
of
Proposition
3.1,
that
M
Θ
∗
arises
[cf.
Proposition
1.2,
(ii)]
from
a
tem-
†
pered
Frobenioid
F
v
—
i.e.,
M
Θ
∗
=
†
M
Θ
∗
(
F
v
)
90
SHINICHI
MOCHIZUKI
—
that
appears
in
a
Θ-Hodge
theater
†
HT
Θ
=
({
†
F
w
}
w∈V
,
†
F
mod
)
[cf.
[IUTchI],
Definition
3.6]
—
cf.,
for
instance,
the
Frobenioid
“F
v
”
of
[IUTchI],
Example
3.2,
(i).
Observe
that
by
applying
the
category-theoretic
constructions
of
Example
3.2,
(i),
(ii),
to
†
F
v
,
one
obtains
data
Π
X
(M
Θ
)
Ψ
=
Ψ
,
Ψ
=
Ψ
;
†
Θ
†
Θ
†
Θ
†
Θ
∞
∞
F
F
,α
F
F
,α
∗
v
v
v
v
Θ
Θ
α∈Π
X
(M
∗
)
α∈Π
X
(M
∗
)
Π
X
(M
Θ
∗
)
Ψ
†
C
v
as
well
as
splittings,
up
to
torsion,
of
each
of
the
monoids
Ψ
†
F
v
Θ
,α
,
∞
Ψ
†
F
v
Θ
,α
.
(i)
(Split
Theta
Monoids)
By
forming
Kummer
classes
relative
to
the
Frobenioid
structure
of
†
F
v
—
i.e.,
in
essence,
by
considering
the
Galois
coho-
mology
classes
that
arise
when
one
extracts
N
-th
roots
of
unity
for
N
∈
N
≥1
[cf.
[FrdII],
Definition
2.1,
(ii);
[IUTchI],
Remark
3.2.3,
(ii);
the
discussion
of
[EtTh],
§5]
—
and
applying
the
description
given
in
Proposition
1.3,
(i),
of
the
exterior
cyclotome
of
a
mono-theta
environment
that
arises
from
a
tempered
Frobenioid,
one
obtains,
for
a
suitable
bijection
of
l
·
Z-torsors
between
[Gal(
Ÿ
v
/Y
v
)-orbits
of
]
“ι”
as
in
Proposition
2.2,
(i),
and
images
of
“α”
via
the
natural
surjection
Π
v
l
·
Z,
collections
of
isomorphisms
of
monoids
Ψ
†
F
v
Θ
,α
∼
→
Ψ
ι
env
(M
Θ
∗
);
∼
ι
Θ
∞
Ψ
env
(M
∗
)
→
∞
Ψ
†
F
v
Θ
,α
—
each
of
which
is
well-defined
up
to
composition
with
an
inner
automorphism
[cf.
the
discussion
of
Example
3.2,
(i)]
and
compatible
with
both
the
respective
conjugation
actions
by
Π
X
(M
Θ
∗
)
and
the
splittings
up
to
torsion
on
the
monoids
under
consideration.
We
shall
denote
these
collections
of
isomorphisms
by
means
of
the
notation
Ψ
†
F
v
Θ
∼
→
Ψ
env
(M
Θ
∗
);
∞
Ψ
†
F
v
Θ
∼
→
Θ
∞
Ψ
env
(M
∗
)
[cf.
the
notation
of
Proposition
3.1,
(i);
Example
3.2,
(i)].
(ii)
(Constant
Monoids)
By
forming
Kummer
classes
relative
to
the
Frobe-
nioid
structure
of
†
F
v
—
i.e.,
in
essence,
by
considering
the
Galois
cohomology
classes
that
arise
when
one
extracts
N
-th
roots
of
unity
for
N
∈
N
≥1
[cf.
[FrdII],
Definition
2.1,
(ii);
[IUTchI],
Remark
3.2.3,
(ii);
[FrdII],
Theorem
2.4]
—
and
ap-
plying
the
description
given
in
Proposition
1.3,
(i),
of
the
exterior
cyclotome
of
a
mono-theta
environment
that
arises
from
a
tempered
Frobenioid,
one
obtains
an
isomorphism
of
monoids
Ψ
†
C
v
∼
→
Ψ
cns
(M
Θ
∗
)
—
which
is
well-defined
up
to
composition
with
an
inner
automorphism
[cf.
the
discussion
of
Example
3.2,
(ii)]
and
compatible
with
the
respective
conjugation
actions
by
Π
X
(M
Θ
∗
).
Proof.
Assertions
(i)
and
(ii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
91
Proposition
3.4.
(Group-theoretic
Theta
Monoids)
Let
†
F
v
be
a
tem-
pered
Frobenioid
as
in
Proposition
3.3.
Consider
the
full
poly-isomorphism
∼
M
Θ
∗
(Π
v
)
†
M
Θ
∗
(
F
v
)
→
—
where
M
Θ
∗
(Π
v
)
is
the
projective
system
of
mono-theta
environments
arising
from
the
algorithm
of
Proposition
1.2,
(i)
[cf.
also
Proposition
1.5,
(i)]
—
of
projective
systems
of
mono-theta
environments.
(i)
(Multiradiality
of
Split
Theta
Monoids)
Each
isomorphism
of
projec-
∼
Θ
†
tive
systems
of
mono-theta
environments
M
Θ
∗
(Π
v
)
→
M
∗
(
F
v
)
induces
compati-
ble
[in
the
evident
sense]
collections
of
isomorphisms
∼
Π
v
→
Π
X
(M
Θ
∗
(Π
v
))
∼
→
†
Π
X
(M
Θ
∗
(
F
v
))
=
†
Π
X
(M
Θ
∗
(
F
v
))
∼
∼
Θ
∞
Ψ
env
(M
∗
(Π
v
))
→
Θ
†
∞
Ψ
env
(M
∗
(
F
v
))
→
Ψ
env
(M
Θ
∗
(Π
v
))
→
∼
†
Ψ
env
(M
Θ
∗
(
F
v
))
→
Ψ
†
F
v
Θ
∼
†
G
v
(M
Θ
∗
(
F
v
))
=
†
G
v
(M
Θ
∗
(
F
v
))
∼
∞
Ψ
†
F
v
Θ
;
and
∼
G
v
→
G
v
(M
Θ
∗
(Π
v
))
→
×
Ψ
env
(M
Θ
∗
(Π
v
))
∼
→
†
×
Ψ
env
(M
Θ
∗
(
F
v
))
∼
→
(Ψ
†
F
v
Θ
)
×
—
where
the
upper
horizontal
isomorphisms
in
each
diagram
are
isomorphisms
of
topological
groups;
the
lower/middle
horizontal
isomorphisms
in
each
diagram
are
isomorphisms
of
[ind-topological]
monoids;
the
lower/middle
horizontal
iso-
morphisms
in
the
first
diagram
are
compatible
with
the
respective
splittings
up
to
torsion;
the
left-hand
square
in
each
diagram
arises
from
the
functoriality
of
the
algorithms
involved,
relative
to
isomorphisms
of
projective
systems
of
mono-theta
environments;
the
right-hand
square
in
each
diagram
arises
from
the
inverses
of
the
isomorphisms
of
the
second
display
of
Proposition
3.3,
(i);
the
superscript
“×”
denotes
the
submonoid
of
units;
the
notation
“G
v
(−)”
is
as
in
Proposition
3.1,
(ii).
Finally,
if
we
write
(Ψ
†
F
v
Θ
)
×μ
for
the
ind-topological
monoid
obtained
by
forming
the
quotient
of
(Ψ
†
F
v
Θ
)
×
by
its
torsion
subgroup,
then
the
functorial
algorithms
Π
v
→
Ψ
env
(M
Θ
∗
(Π
v
));
Π
v
→
∞
Ψ
env
(M
Θ
∗
(Π
v
))
Θ
—
where
we
think
of
Ψ
env
(M
Θ
∗
(Π
v
)),
∞
Ψ
env
(M
∗
(Π
v
))
as
being
equipped
with
their
natural
Π
v
-actions
and
splittings
up
to
torsion
[cf.
Proposition
3.1,
(i)]
—
obtained
by
composing
the
algorithms
of
Propositions
1.2,
(i);
3.1,
(i),
are
compatible,
relative
to
the
above
displayed
diagrams,
with
arbitrary
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-topological
monoid
equipped
with
the
action
of
a
topological
group,
determined
by]
the
pair
†
G
v
(M
Θ
∗
(
F
v
))
(Ψ
†
F
v
Θ
)
×μ
92
SHINICHI
MOCHIZUKI
which
arise
as
Ism-multiples
of
automorphisms
induced
by
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-topological
monoid
equipped
with
the
action
†
×
[cf.
of
a
topological
group,
determined
by]
the
pair
G
v
(M
Θ
∗
(
F
v
))
(Ψ
†
F
v
Θ
)
Example
1.8,
(iv);
Remark
1.8.1;
Remark
1.11.1,
(i),
(b)]
—
in
the
sense
that
the
natural
functor
“Ψ
R
”
of
Corollary
1.12,
(iii),
is
multiradially
defined.
(ii)
(Uniradiality
of
Constant
Monoids)
Each
isomorphism
of
projective
∼
Θ
†
systems
of
mono-theta
environments
M
Θ
∗
(Π
v
)
→
M
∗
(
F
v
)
induces
compatible
collections
of
isomorphisms
∼
Π
v
→
Π
X
(M
Θ
∗
(Π
v
))
∼
→
†
Π
X
(M
Θ
∗
(
F
v
))
=
†
Π
X
(M
Θ
∗
(
F
v
))
Ψ
cns
(M
Θ
∗
(Π
v
))
→
∼
†
Ψ
cns
(M
Θ
∗
(
F
v
))
→
∼
Ψ
†
C
v
∼
→
∼
†
G
v
(M
Θ
∗
(
F
v
))
=
†
G
v
(M
Θ
∗
(
F
v
))
and
G
v
→
G
v
(M
Θ
∗
(Π
v
))
×
Ψ
cns
(M
Θ
∗
(Π
v
))
∼
→
†
×
Ψ
cns
(M
Θ
∗
(
F
v
))
∼
→
(Ψ
†
C
v
)
×
—
where
the
upper
horizontal
isomorphisms
in
each
diagram
are
isomorphisms
of
topological
groups;
the
lower
horizontal
isomorphisms
in
each
diagram
are
isomor-
phisms
of
[ind-topological]
monoids;
the
second
diagram
may
be
naturally
iden-
tified
with
the
second
displayed
commutative
diagram
of
(i);
the
left-hand
square
in
each
diagram
arises
from
the
functoriality
of
the
algorithms
involved,
relative
to
isomorphisms
of
projective
systems
of
mono-theta
environments;
the
right-hand
square
in
each
diagram
arises
from
the
inverse
of
the
displayed
isomorphism
of
Proposition
3.3,
(ii);
the
superscript
“×”
denotes
the
submonoid
of
units;
the
no-
tation
“G
v
(−)”
is
as
in
Proposition
3.1,
(ii).
Finally,
if
we
write
(Ψ
†
C
v
)
×μ
for
the
ind-topological
monoid
obtained
by
forming
the
quotient
of
(Ψ
†
C
v
)
×
by
its
torsion
subgroup,
then
the
functorial
algorithm
Π
v
→
Ψ
cns
(M
Θ
∗
(Π
v
))
—
where
we
think
of
Ψ
cns
(M
Θ
∗
(Π
v
))
as
being
equipped
with
its
natural
Π
v
-action
[cf.
Proposition
3.1,
(ii)]
—
obtained
by
composing
the
algorithms
of
Proposition
1.2,
(i);
3.1,
(ii),
depends
on
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.11,
(b)
[cf.
Remark
1.11.5,
(ii);
the
use
of
the
surjection
of
Remark
1.11.5,
(i),
in
the
algorithm
of
Proposition
3.1,
(ii)],
hence
fails
to
be
compatible,
relative
to
the
above
displayed
diagrams,
with
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-topological
monoid
equipped
with
the
action
of
a
topological
group,
determined
by]
the
pair
†
G
v
(M
Θ
∗
(
F
v
))
(Ψ
†
C
v
)
×μ
which
arise
from
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-
topological
monoid
equipped
with
the
action
of
a
topological
group,
determined
by]
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
93
†
×
the
pair
G
v
(M
Θ
[cf.
Remarks
1.11.1,
(i),
(b);
1.8.1]
—
in
∗
(
F
v
))
(Ψ
†
C
v
)
the
sense
that
this
algorithm,
as
given,
only
admits
a
uniradial
formulation
[cf.
Remarks
1.11.3,
(iv);
1.11.5,
(ii)].
Proof.
Assertions
(i)
and
(ii)
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.4.1.
(i)
Note
that
the
pairs
†
“G
v
(M
Θ
∗
(
F
v
))
(Ψ
†
F
v
Θ
)
×μ
”
and
†
“G
v
(M
Θ
∗
(
F
v
))
(Ψ
†
C
v
)
×μ
”
that
appear
in
Proposition
3.4,
(i),
(ii),
correspond
to
the
pair
“G
O
×μ
(G)”
that
appears
in
the
discussion
of
Remark
1.11.3,
(ii)
—
i.e.,
the
data
that
arises
by
replacing
the
“O
×
”
that
appears
in
the
Θ-link
of
[IUTchI],
Corollary
3.7,
(iii),
by
“O
×μ
”.
That
is
to
say,
from
the
point
of
view
of
the
present
series
of
papers,
the
significance
of
Proposition
3.4
lies
in
the
point
of
view
that
the
multiradiality
(respectively,
uniradiality)
asserted
in
Proposition
3.4,
(i)
(respectively,
(ii)),
may
be
thought
of
as
a
statement
of
the
com-
patibility
(respectively,
incompatibility)
of
the
algorithm
in
question
with
the
“O
×μ
-version”
of
the
Θ-link
of
[IUTchI],
Corollary
3.7,
(iii).
(ii)
One
important
consequence
of
the
theory
to
be
developed
in
[IUTchIII]
[cf.
Remark
2.9.1,
(iii)]
is
the
result
that,
by
applying
the
theory
of
log-shells
[cf.
[AbsTopIII]],
one
may
construct
certain
algorithms
related
to
the
algorithm
of
Proposition
3.4,
(ii),
that
[yield
functors
which]
are
manifestly
multiradially
defined
—
albeit
at
the
cost
of
allowing
for
certain
[relatively
mild!]
indeterminacies.
The
following
two
corollaries
will
play
a
fundamental
role
in
the
present
series
of
papers.
Corollary
3.5.
(Mono-theta-theoretic
Gaussian
Monoids)
Let
M
Θ
∗
be
as
in
Proposition
3.1
[cf.
also
Corollary
2.8,
in
the
case
where
γ
=
1;
Remark
3.5.1
below].
For
t
∈
LabCusp
±
(Π
X
(M
Θ
∗
)),
we
shall
denote
copies
labeled
by
t
of
various
objects
functorially
constructed
from
M
Θ
∗
by
means
of
a
subscript
“t”.
Also,
we
shall
write
Π
X
(M
Θ
⊆
Π
X
(M
Θ
⊆
Π
C
(M
Θ
∗
)
∗
)
∗
)
Δ
X
(M
Θ
∗
)
⊆
Δ
X
(M
Θ
∗
)
⊆
Δ
C
(M
Θ
∗
)
for
the
inclusions
—
which
may
be
functorially
constructed
from
Π
X
(M
Θ
∗
)
—
cor-
cor
±
cor
responding
to
the
inclusions
Π
v
⊆
Π
±
of
Definition
2.3,
v
⊆
Π
v
,
Δ
v
⊆
Δ
v
⊆
Δ
v
(i).
94
SHINICHI
MOCHIZUKI
(i)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
If
we
think
of
the
cuspidal
inertia
groups
⊆
Π
X
(M
Θ
∗
)
corresponding
to
t
as
subgroups
Θ
of
cuspidal
inertia
groups
of
Π
X
(M
∗
)
[cf.
Remark
2.3.1],
then
the
Δ
X
(M
Θ
∗
)-outer
±
∼
Θ
Θ
Θ
action
of
F
l
=
Δ
C
(M
∗
)/Δ
X
(M
∗
)
on
Π
X
(M
∗
)
[cf.
Corollary
2.4,
(iii);
Remark
1.1.1,
(iv),
or,
alternatively,
when
applicable,
Proposition
1.3,
(ii),
(iii)]
induces
isomorphisms
between
the
pairs
Θ
G
v
(M
Θ
¨
)
t
Ψ
cns
(M
∗
)
t
∗
—
consisting
of
a
labeled
ind-topological
monoid
equipped
with
the
action
of
a
labeled
topological
group
[cf.
Proposition
3.1,
(ii)]
—
for
distinct
t
∈
LabCusp
±
±
(Π
X
(M
Θ
∗
)).
We
shall
refer
to
these
isomorphisms
as
[F
l
-]symmetrizing
iso-
morphisms
[cf.
Remark
3.5.2
below].
We
shall
denote
by
means
of
a
subscript
“|t|
∈
|F
l
|”
the
result
of
identifying
copies
labeled
by
t,
−t
via
a
suitable
sym-
metrizing
isomorphism.
We
shall
denote
by
means
of
a
subscript
“|F
l
|”
(respec-
tively,
“F
l
”)
the
diagonal
embedding,
determined
by
suitable
symmetrizing
isomorphisms,
inside
the
direct
product
of
copies
labeled
by
|t|
∈
|F
l
|
(respectively,
Θ
|t|
∈
F
l
).
In
particular,
by
restricting
the
monoid
Ψ
cns
(M
∗
)
of
Proposition
3.1,
δ
”]
described
in
detail
(ii),
via
the
restriction
operations
[i.e.,
to
“Π
M
Θ
¨
”
and
“D
t,μ
−
∗
in
Corollary
2.8,
(i),
(ii),
one
obtains
a
collection
of
compatible
morphisms
Θ
)
←
Π
v
G
v
(M
Θ
Π
X
(M
Θ
¨
(M
∗
∗
¨
)
¨
)
|F
l
|
∗
∼
→
Ψ
cns
(M
Θ
∗
)
Ψ
cns
(M
Θ
∗
)
|F
l
|
—
where
the
notation
“”
denotes
the
natural
actions;
the
bottom
horizontal
arrow
is
an
isomorphism
of
monoids
—
which
are
compatible
with
the
various
sym-
metrizing
isomorphisms
and
well-defined
up
to
composition
with
an
inner
automorphism
of
Π
X
(M
Θ
∗
)
[i.e.,
up
to
composition
with
the
conjugation
action
Θ
Θ
by
Π
X
(M
Θ
¨
(M
∗
∗
)
on
the
pair
Π
v
¨
)
Ψ
cns
(M
∗
)].
Put
another
way,
this
inner
automorphism
indeterminacy
—
which,
a
priori,
depends
on
the
index
|t|
—
is,
in
fact,
independent
of
|t|
∈
|F
l
|.
(ii)
(Gaussian
Monoids)
We
shall
refer
to
an
element
of
the
set
def
|t|
Θ
l
(M
Θ
)
=
θ
(M
)
⊆
Ψ
cns
(M
Θ
θ
F
env
¨
¨
∗
)
|t|
∗
∗
env
|t|∈F
l
|t|∈F
l
[cf.
the
notation
of
Corollary
2.8,
(i),
(ii)]
—
which
is
of
cardinality
(2l)
l
—
as
a
value-profile.
Then
by
applying
[the
various
objects
constructed
from]
the
symmetrizing
isomorphisms
of
(i),
together
with
the
functorial
algorithm
[for
ι
Θ
restricting
elements
of
θ
ι
env
(M
Θ
∗
),
∞
θ
env
(M
∗
)]
of
Corollary
2.8,
(i),
(ii),
one
obtains
a
functorial
algorithm
for
constructing
two
collections
of
submonoids
M
Θ
∗
→
def
Ψ
gau
(M
Θ
∗
)
=
def
Θ
∞
Ψ
gau
(M
∗
)
=
def
Ψ
ξ
(M
Θ
∗
)
=
Θ
N
Ψ
×
cns
(M
∗
)
F
·
ξ
l
def
Θ
∞
Ψ
ξ
(M
∗
)
=
⊆
|t|∈F
l
Θ
Q
≥0
Ψ
×
cns
(M
∗
)
F
·
ξ
l
⊆
Ψ
cns
(M
Θ
∗
)
|t|
|t|∈F
l
ξ
,
Ψ
cns
(M
Θ
∗
)
|t|
ξ
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
95
—
where
the
superscript
“×”
denotes
the
submonoid
of
units;
ξ
ranges
over
the
value-profiles;
“ξ
Q
≥0
”
denotes
the
submonoid
generated
by
the
N
-th
roots
[for
N
∈
N
≥1
]
of
ξ
[which
are
uniquely
determined,
up
to
multiplication
by
an
ele-
Θ
ment
of
the
N
-torsion
subgroup
of
Ψ
×
cns
(M
∗
)
F
!]
that
arise
by
restricting
elements
l
Θ
Θ
of
∞
θ
ι
env
(M
Θ
∗
);
each
Ψ
ξ
(M
∗
)
is
equipped
with
a
natural
action
by
G
v
(M
∗
¨
)
F
.
l
Θ
We
shall
refer
to
each
Ψ
ξ
(M
Θ
∗
)
or
∞
Ψ
ξ
(M
∗
)
as
a
Gaussian
monoid.
Here,
the
Θ
×
Θ
2l·N
submonoid
Ψ
2l·ξ
(M
Θ
is
indepen-
∗
)
⊆
Ψ
ξ
(M
∗
)
generated
by
Ψ
cns
(M
∗
)
F
and
ξ
l
dent
of
the
value-profile
ξ.
Finally,
the
restriction
operations
described
in
detail
in
Corollary
2.8,
(i),
(ii),
determine
a
collection
of
compatible
[in
the
evident
sense]
morphisms
[cf.
Remark
3.6.1
below]
Θ
Π
X
(M
Θ
)
←
Π
v
{G
v
(M
Θ
¨
(M
∗
∗
¨
)
¨
)
|t|
}
|t|∈F
∗
l
∼
ι
Θ
∞
Ψ
env
(M
∗
)
→
Ψ
ι
env
(M
Θ
∗
)
→
∼
Θ
∞
Ψ
ξ
(M
∗
)
Ψ
ξ
(M
Θ
∗
)
—
where
the
“”
in
the
first
line
denotes
the
compatibility
of
the
action
[de-
noted
by
the
second
“”
in
the
second
line]
of
G
v
(M
Θ
¨
)
|t|
on
the
factor
labeled
∗
Θ
“|t|”
of
the
direct
product
containing
∞
Ψ
ξ
(M
∗
)
[cf.
the
definition
of
∞
Ψ
ξ
(M
Θ
∗
)]
Θ
with
the
inclusions
G
v
(M
Θ
)
→
Π
(M
)
determined
by
the
various
choices
of
¨
v
∗
¨
∗
δ
the
“D
t,μ
−
”
[cf.
Corollary
2.8,
(i),
(ii)]
that
gave
rise
to
the
value-profile
ξ;
the
first
“”
in
the
second
line
denotes
the
natural
action;
the
lower/middle
horizontal
arrows
are
isomorphisms
of
monoids
—
which
is
well-defined
up
to
composition
with
a(n)
[single!]
inner
automorphism
of
Π
X
(M
Θ
∗
)
and
compatible
[in
the
ev-
Θ
ident
sense]
with
the
equalities
of
submonoids
Ψ
2l·ξ
1
(M
Θ
∗
)
=
Ψ
2l·ξ
2
(M
∗
)
for
distinct
value-profiles
ξ
1
,
ξ
2
.
For
simplicity,
we
shall
use
the
notation
Ψ
env
(M
Θ
∗
)
∼
→
Ψ
gau
(M
Θ
∗
);
Θ
∞
Ψ
env
(M
∗
)
∼
→
Θ
∞
Ψ
gau
(M
∗
)
to
denote
these
collections
of
compatible
morphisms
induced
by
restriction.
(iii)
(Constant
Monoids
and
Splittings)
Denote
the
zero
element
of
|F
l
|
by
0
∈
|F
l
|.
Then
[in
the
notation
of
(i)]
the
diagonal
submonoid
Ψ
cns
(M
Θ
∗
)
|F
l
|
determines
—
i.e.,
may
be
thought
of
as
the
graph
of
—
an
isomorphism
of
monoids
∼
→
Ψ
cns
(M
Θ
Ψ
cns
(M
Θ
∗
)
0
∗
)
F
l
that
is
compatible
with
the
respective
labeled
G
v
(M
Θ
¨
)-actions.
∗
Moreover,
the
restriction
operations
to
zero-labeled
evaluation
points
described
in
detail
in
Corollary
2.8,
(i),
(ii),
(iii),
determine
a
splitting
up
to
torsion
of
each
of
the
Gaussian
monoids
×
Θ
N
Ψ
ξ
(M
Θ
∗
)
=
Ψ
cns
(M
∗
)
F
·
ξ
,
l
Θ
∞
Ψ
ξ
(M
∗
)
Θ
Q
≥0
=
Ψ
×
cns
(M
∗
)
F
·
ξ
l
Θ
[cf.
the
definition
of
Ψ
ξ
(M
Θ
∗
),
∞
Ψ
ξ
(M
∗
)
in
(ii)]
which
is
compatible,
relative
to
the
restriction
isomorphisms
of
the
third
display
of
(ii),
with
the
splittings
up
to
torsion
of
Proposition
3.1,
(i).
96
SHINICHI
MOCHIZUKI
Proof.
The
various
assertions
of
Corollary
3.5
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.5.1.
(i)
Note
that
in
Corollary
3.5,
unlike
the
situation
of
Corollary
2.8,
we
took
γ
to
be
=
1.
This
was
done
primarily
to
simplify
the
notation
and
does
not
result
in
any
substantive
loss
of
generality.
Indeed,
one
may
always
simply
take
the
“M
Θ
∗
”
of
Θ
γ
Corollary
3.5
to
be
the
“(M
∗
)
”
of
Corollary
2.8.
Alternatively,
one
may
observe
δ
”
that
occurs
in
the
various
restriction
that
the
“δ”
that
appears
in
the
“D
t,μ
−
operations
invoked
in
Corollary
3.5
[cf.
Corollary
2.8,
(i),
(ii)]
is
arbitrary,
i.e.,
it
is
subject
to
the
independent
conjugation
indeterminacies
discussed
in
Corollary
2.5,
(iii);
Remark
2.5.2.
(ii)
In
the
present
context,
it
is
useful
to
recall
that
from
the
point
of
view
of
the
discussion
of
[IUTchI],
Remark
3.2.3,
(i),
the
various
Π
X
(M
Θ
∗
)-conjugacy
indeterminacies
that
appear
in
Corollary
3.5
are
applied,
in
the
context
of
the
theory
of
the
present
series
of
papers,
to
identify
the
various
Π
X
(M
Θ
∗
)-conjugates
Θ
of
Π
v
¨
(M
∗
¨
)
[or,
alternatively,
“ι’s”]
with
one
another.
Remark
3.5.2.
Before
proceeding,
it
is
useful
to
pause
to
consider
the
significance
of
the
symmetrizing
isomorphisms
of
Corollary
3.5,
(i).
(i)
We
begin
by
discussing
a
simple
combinatorial
model
of
the
phenomenon
of
interest.
Consider
the
totally
ordered
set
E
=
{0,
1}
whose
ordering
is
completely
determined
by
the
inequality
0
<
1
—
which
we
shall
denote,
in
the
following
discussion,
by
the
notation
“≺”.
Then
one
may
consider
labeled
copies
≺
0
,
≺
1
of
≺.
Now
suppose
that
one
attempts
to
identify
these
labeled
copies
≺
0
,
≺
1
by
simply
forgetting
the
labels.
This
amounts,
in
effect,
to
sending
the
two
distinct
subscripted
labels
E
0,
1
→
∗
to
a
single
point
“∗”.
In
particular,
this
naive
approach
to
identifying
the
labeled
copies
≺
0
,
≺
1
fails
to
be
compatible
—
in
a
sense
that
we
shall
examine
in
more
detail
in
the
discussion
to
follow
—
with
operations
that
require
one
to
distinguish
the
two
labels
0,
1
∈
E.
Now
if,
to
avoid
confusion,
one
writes
S
for
the
underlying
set
of
E
[i.e.,
obtained
from
E
by
forgetting
the
ordering
on
E],
then
one
has
a
natural
Aut(S)-orbit
of
bijections
∼
E
→
S
Aut(S)
—
where
Aut(S)
∼
=
Z/2Z.
Next,
let
us
suppose
that
we
are
given
an
object
F
(≺)
functorially
constructed
from
[the
“totally
ordered
set
of
cardinality
two”]
≺.
Then
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
97
any
“factorization”
of
the
functorial
construction
F
(−)
[i.e.,
on
“totally
ordered
sets
of
cardinality
two”]
through
a
functorial
construction
F
sym
(S)
Aut(S)
on
unordered
sets
of
cardinality
two
[i.e.,
relative
to
the
“forgetful
functor”
that
associates
to
an
ordered
set
the
underlying
unordered
set]
may
be
thought
of
as
a
collection
of
“symmetrizing
isomorphisms”
[cf.
the
discussion
of
(ii)
below;
Corollary
3.5,
(i)],
or,
alternatively,
as
“descent
data”
for
F
(−)
from
E
to
the
“orbiset
quotient”
of
S
by
Aut(S).
Moreover,
this
“descent
data”
satisfies
the
crucial
property
that
it
allows
one
to
perform
this
“descent
to
the
orbiset
quotient”
in
such
a
way
that
one
is
never
required
to
violate
the
bijective
relationship
—
albeit
via
an
in-
determinate
bijection!
—
between
E
and
S.
By
contrast,
the
“naive
approach”
discussed
above
may
be
thought
of
as
corre-
sponding
to
working
with
the
“coarse
set-theoretic
quotient”
Q
of
S
by
Aut(S)
def
—
which
we
shall
think
of
as
consisting
of
a
single
point
∗
=
{0,
1}
∈
Q
=
{∗}.
def
Now
suppose,
for
instance,
in
the
case
F
(≺)
=
≺,
that
one
attempts
to
regard
def
F
(≺)
(−)
=
≺
(−)
[where
(−)
∈
S]
as
an
object
“pulled
back”
from
a
copy
≺
Q
[i.e.,
“0
Q
<
1
Q
”]
of
≺
over
Q.
On
the
other
hand,
if
one
wishes
to
relate
each
point
def
s
∈
S
to
one
or
more
points
∈
E
Q
=
{0
Q
,
1
Q
}
via
an
Aut(S)-equivariant
assign-
ment
in
such
a
way
that
every
point
of
E
Q
appears
in
the
image
of
this
assignment,
then
one
has
no
choice
but
to
assign
to
each
point
s
∈
S
the
collection
of
all
points
∈
E
Q
.
Put
another
way,
one
must
contend
with
an
independent
indeterminacy
s
→
0
Q
?
1
Q
?
for
each
s
∈
S
—
i.e.,
if
we
write
S
=
{0
S
,
1
S
},
then
these
indeterminacies
give
rise
to
a
total
of
4
possibilities
0
S
→
0
Q
?
1
Q
?
1
S
→
0
Q
?
1
Q
?
for
the
desired
assignment,
certain
of
which
[i.e.,
0
S
,
1
S
→
0
Q
and
0
S
,
1
S
→
1
Q
]
fail
to
be
bijective.
Here,
it
is
useful
to
note
that
to
synchronize
these
indeterminacies
amounts,
tautologically,
to
the
requirement
of
an
“automorphism
of
≺
Q
that
induces
the
unique
nontrivial
automorphism
of
the
set
E
Q
=
{0
Q
,
1
Q
}”.
On
the
other
hand,
by
the
definition
of
an
“inequality”,
it
is
a
tautology
that
such
an
automorphism
of
≺
Q
cannot
exist.
Finally,
in
this
context,
it
is
useful
to
recall
that
this
difference
between
“crushing
the
set
E
to
a
single
point”
and
“symmetrizing
without
violating
the
bijective
relationship
to
E”
is
precisely
the
topic
of
the
discussion
of
[IUTchI],
Remark
4.9.2,
(i);
[IUTchI],
Remark
6.12.4,
(i)
—
cf.,
especially,
[IUTchI],
Fig.
4.5.
(ii)
The
starting
point
of
the
theory
surrounding
the
symmetrizing
isomor-
phisms
of
Corollary
3.5,
(i),
is
the
connectedness
—
or
“single
basepoint”
—
observed
in
the
discussion
of
Remark
2.6.1,
(i),
together
with
the
compatibility
of
this
connectedness
with
a
certain
F
±
l
-symmetry,
as
discussed
in
Remark
2.6.2,
98
SHINICHI
MOCHIZUKI
(i).
These
symmetrizing
isomorphisms
may
be
applied
to
labeled
copies
of
vari-
Θ
Θ
Θ
ous
objects
constructed
from
M
Θ
∗
—
e.g.,
Ψ
cns
(M
∗
),
G
v
(M
∗
),
Π
μ
(M
∗
)
—
cf.
the
discussion
of
“conjugate
synchronization”
in
Remark
2.6.1,
(i).
Note
that
in
the
absence
of
the
F
±
l
-symmetry
involved,
the
“single
basepoint”
under
consideration
has
a
rigidifying
effect
not
only
on
the
various
conjugates
involved,
but
also
on
the
labels
under
consideration.
That
is
to
say,
a
priori,
it
is
quite
possible
that
the
desired
rigidity
of
the
conjugates
involved
depends
on
the
rigidity
of
the
labels
under
consideration.
Indeed,
this
is
precisely
what
happens
when
the
data
that
one
wishes
to
synchronize
—
i.e.,
such
as
monoids,
absolute
Galois
groups,
or
cyclotomes
—
consists,
for
instance,
of
an
arrow
from
one
label
to
another,
as
was
[essentially]
the
case
in
the
discussion
of
the
combinatorial
model
of
(i).
Put
another
way,
the
significance
of
the
F
±
l
-symmetry
under
consideration
lies
precisely
in
the
observation
that
this
symmetry
serves
to
eliminate
this
unwanted
“a
priori”
possibility.
This
is
in
some
sense
the
central
principle
illustrated
by
the
combinatorial
model
of
(i).
Put
in
other
words,
this
“central
principle”
discussed
in
(i)
may
be
sum-
marized,
in
the
situation
of
Corollary
3.5,
as
follows:
the
F
±
l
-symmetry
under
consideration
allows
one
to
construct
(a)
symmetrizing
isomorphisms
[cf.
Corollary
3.5,
(i)]
in
a
fashion
that
is
compatible
with
maintaining
a
(b)
bijective
link
with
the
set
of
labels
LabCusp
±
(Π
X
(M
Θ
∗
))
—
which
is
necessary
in
order
to
construct
the
Gaussian
monoids
[i.e.,
which
involve
distinct
values
at
distinct
labels!]
in
Corollary
3.5,
(ii)
—
all
relative
to
(c)
a
single
basepoint
[i.e.,
which
gives
rise
to
the
single
topological
group
Π
X
(M
Θ
∗
)
—
cf.
the
discussion
of
Remark
2.6.2,
(i)]
—
which
is
necessary
in
order
to
establish
conjugate
synchronization.
(iii)
In
the
context
of
Corollary
3.5,
(i),
one
essential
aspect
of
the
F
±
l
-
Θ
symmetry
under
consideration
is
that
this
symmetry
arises
from
a
Δ
X
(M
∗
)-outer
±
Θ
∼
action
of
Δ
C
(M
Θ
[cf.
the
discussion
of
Remark
2.6.2,
(i)].
That
∗
)/Δ
X
(M
∗
)
→
F
l
is
to
say,
the
fact
that
this
action
may
be
formulated
entirely
in
terms
of
conju-
gation
by
elements
of
geometric
[i.e.,
“Δ”]
fundamental
groups
—
that
is
to
say,
as
opposed
to
arithmetic
[i.e.,
“Π”]
fundamental
groups
—
plays
a
crucial
role
in
establishing
the
conjugate
synchronization
of
the
various
copies
of
“G
v
(M
Θ
∗
)”
Θ
[and
objects
constructed
from
“G
v
(M
∗
)”]
under
consideration
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.6,
(ii)].
(iv)
If
one
thinks
of
the
F
±
l
-symmetries
that
appear
in
the
conjugate
synchro-
nization
of
Corollary
3.5,
(i),
as
“connecting”
the
various
copies
of
objects
at
distinct
evaluation
points,
then
it
is
perhaps
natural
to
regard
the
“conjugate
synchro-
nization
via
symmetry”
of
Corollary
3.5,
(i),
as
a
sort
of
nonarchimedean
version
of
the
“conjugate
synchronization
via
connectedness”
discussed
in
Remark
2.6.1,
(i),
which
may
be
thought
of
as
being
based
on
the
“archimedean”
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
99
connectedness
of
the
subgraph
Γ
X
⊆
Γ
X
[cf.
the
discussion
of
Remarks
2.6.1,
(i);
2.8.3].
(v)
In
§4
below,
we
shall
generalize
the
ideas
discussed
in
the
present
Remark
3.5.2
concerning
conjugate
synchronization
in
the
case
of
v
∈
V
bad
to
the
global
portion,
as
well
as
to
the
portion
at
good
v
∈
V
good
,
of
a
D-Θ
±ell
-Hodge
theater
[cf.
the
discussions
of
Remark
2.6.2,
(i);
Remark
3.8.2
below].
Remark
3.5.3.
The
delicacy
and
subtlety
of
the
theory
surrounding
Corollary
3.5,
(i),
may
be
thought
of
as
a
consequence
of
the
requirement
of
simultaneously
satisfying
the
conditions
(a),
(b),
(c)
discussed
in
Remark
3.5.2,
(ii).
On
the
other
hand,
if
one
is
willing
to
eliminate
condition
(c)
from
one’s
arguments,
then
one
may
obtain
symmetrizing
isomorphisms
by
simply
applying
the
functors
of
[IUTchI],
Proposition
6.8,
(i),
(ii),
(iii);
[IUTchI],
Proposition
6.9,
(i),
(ii)
—
i.e.,
by
passing
to
D-Θ
ell
-bridges
or
[holomorphic
or
mono-analytic]
capsules
or
processions.
Here,
we
observe
that
this
“multi-basepoint”
approach
to
constructing
symmetrizing
isomorphisms
is
compatible
with
the
single
basepoint
F
±
l
-symmetric
approach
of
Corollary
3.5,
(i),
relative
to
the
evident
“forgetful
functors”.
We
leave
the
routine
details
to
the
reader.
Corollary
3.6.
(Frobenioid-theoretic
Gaussian
Monoids)
Suppose
that
we
are
in
the
situation
of
Proposition
3.3,
i.e.,
that
M
Θ
∗
†
M
Θ
∗
(
F
v
)
=
—
where
†
F
v
is
a
tempered
Frobenioid.
We
continue
to
use
the
conventions
introduced
in
Corollary
3.5
concerning
subscripted
labels.
(i)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
The
isomorphism
of
Proposition
3.3,
(ii)
[or,
alternatively,
Proposition
1.3,
(ii),
(iii)],
determines,
for
each
t
∈
LabCusp
±
(Π
X
(M
Θ
∗
)),
a
collection
of
compatible
mor-
phisms
∼
Π
X
(M
Θ
)
G
v
(M
Θ
→
G
v
(M
Θ
∗
t
∗
)
t
¨
)
t
∗
(Ψ
†
C
v
)
t
∼
→
Ψ
cns
(M
Θ
∗
)
t
—
which
are
well-defined
up
to
composition
with
an
inner
automorphism
of
±
±
Θ
Π
X
(M
Θ
∗
)
which
is
independent
of
t
∈
LabCusp
(Π
X
(M
∗
))
—
as
well
as
[F
l
-
]symmetrizing
isomorphisms,
induced
by
the
Δ
X
(M
Θ
)-outer
action
of
F
±
∼
=
∗
l
Θ
Θ
Δ
C
(M
Θ
∗
)/Δ
X
(M
∗
)
on
Π
X
(M
∗
)
[cf.
Corollary
3.5,
(i);
Remark
1.1.1,
(iv),
or,
alternatively,
Proposition
1.3,
(ii),
(iii)],
between
the
data
indexed
by
distinct
t
∈
LabCusp
±
(Π
X
(M
Θ
∗
)).
(ii)
(Gaussian
Monoids)
For
each
value-profile
ξ
[cf.
Corollary
3.5,
(ii)],
write
(Ψ
†
C
v
)
|t|
Ψ
F
ξ
(
†
F
v
)
⊆
∞
Ψ
F
ξ
(
†
F
v
)
⊆
|t|∈F
l
100
SHINICHI
MOCHIZUKI
∼
for
the
submonoids
determined,
respectively,
via
the
isomorphisms
(Ψ
†
C
v
)
|t|
→
Θ
Θ
Ψ
cns
(M
Θ
∗
)
|t|
of
(i),
by
the
monoids
Ψ
ξ
(M
∗
),
∞
Ψ
ξ
(M
∗
)
of
Corollary
3.5,
(ii),
and
Ψ
F
gau
(
†
F
v
)
=
def
Ψ
F
ξ
(
†
F
v
)
ξ
†
∞
Ψ
F
gau
(
F
v
)
=
def
,
†
∞
Ψ
F
ξ
(
F
v
)
ξ
—
where
ξ
ranges
over
the
value-profiles.
Thus,
each
monoid
Ψ
F
ξ
(
†
F
v
)
is
equipped
with
a
natural
action
by
G
v
(M
Θ
∗
)
F
.
Then
by
composing
the
Kummer
isomor-
l
phisms
discussed
in
(i)
above
and
Proposition
3.3,
(i),
(ii),
with
the
restriction
isomorphisms
of
Corollary
3.5,
(ii),
one
obtains
a
diagram
of
compatible
mor-
phisms
Θ
Π
v
¨
(M
∗
¨
)
Θ
Π
v
¨
(M
∗
¨
)
=
∼
∼
→
ι
Θ
∞
Ψ
env
(M
∗
)
→
Ψ
†
F
v
Θ
,α
→
∼
Ψ
ι
env
(M
Θ
∗
)
→
l
∼
→
{G
v
(M
Θ
∗
)
|t|
}
|t|∈F
l
∞
Ψ
†
F
v
Θ
,α
{G
v
(M
Θ
¨
)
|t|
}
|t|∈F
∗
∼
Θ
∞
Ψ
ξ
(M
∗
)
→
†
∞
Ψ
F
ξ
(
F
v
)
Ψ
ξ
(M
Θ
∗
)
→
∼
Ψ
F
ξ
(
†
F
v
)
∼
—
where
the
“”
in
the
first
line
[cf.
also
the
second
and
third
“”
in
the
sec-
Θ
ond
line]
is
as
in
Corollary
3.5,
(ii);
we
recall
the
natural
inclusion
Π
v
¨
(M
∗
¨
)
→
Θ
Π
X
(M
∗
)
—
which
is
well-defined
up
to
composition
with
a(n)
[single!]
inner
automorphism
of
Π
X
(M
Θ
∗
)
and
compatible
[in
the
evident
sense]
with
the
equal-
ities
of
submonoids
involving
“Ψ
2l·ξ
(−)”
[cf.
Corollary
3.5,
(ii)].
For
simplicity,
we
shall
use
the
notation
Ψ
†
F
v
Θ
∞
Ψ
†
F
v
Θ
∼
→
∼
→
∼
Ψ
env
(M
Θ
∗
)
→
Θ
∞
Ψ
env
(M
∗
)
→
∼
∼
Ψ
F
gau
(
†
F
v
);
→
Ψ
gau
(M
Θ
∗
)
Θ
∞
Ψ
gau
(M
∗
)
∼
→
†
∞
Ψ
F
gau
(
F
v
)
to
denote
these
collections
of
compatible
morphisms.
(iii)
(Constant
Monoids
and
Splittings)
Relative
to
the
notational
con-
ventions
adopted
thus
far
[cf.
also
Corollary
3.5,
(iii)],
the
diagonal
submonoid
(Ψ
†
C
v
)
|F
l
|
determines
—
i.e.,
may
be
thought
of
as
the
graph
of
—
an
isomor-
phism
of
monoids
∼
(Ψ
†
C
v
)
0
→
(Ψ
†
C
v
)
F
l
that
is
compatible
with
the
respective
labeled
G
v
(M
Θ
∗
)-actions.
Moreover,
the
splittings
of
Corollary
3.5,
(iii),
determine
splittings
up
to
torsion
of
each
of
the
[“Frobenioid-theoretic”]
Gaussian
monoids
N
Ψ
F
ξ
(
†
F
v
)
=
(Ψ
×
†
C
)
F
·
Im(ξ)
,
v
l
†
∞
Ψ
F
ξ
(
F
v
)
Q
≥0
=
(Ψ
×
†
C
)
F
·
Im(ξ)
v
l
—
where
“Im(ξ)”
denotes
the
image
of
ξ
via
the
isomorphisms
discussed
in
(ii)
—
which
are
compatible,
relative
to
the
various
isomorphisms
of
the
third
display
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
101
of
(ii),
with
the
splittings
up
to
torsion
of
Proposition
3.1,
(i);
Proposition
3.3,
(i);
Corollary
3.5,
(iii).
Proof.
The
various
assertions
of
Corollary
3.6
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.6.1.
The
“Galois
compatibility”
denoted
by
the
“”
in
the
third
display
of
Corollaries
3.5,
(ii);
3.6,
(ii)
—
involving
the
monoids
“
∞
Ψ”
[i.e.,
not
just
the
monoids
“Ψ”!]
—
corresponds
precisely
to
the
“Galois
functoriality”
[cf.
Fig.
1.5]
of
the
discussion
of
Remark
1.12.4.
Remark
3.6.2.
The
diagram
in
the
third
display
of
Corollary
3.6,
(ii)
—
which
may
be
thought
of
as
a
sort
of
concrete
realization
of
the
principle
of
Galois
evaluation
discussed
in
Remark
1.12.4
[cf.
also
Remark
3.6.1]
—
will
play
a
central
role
in
the
theory
of
the
present
series
of
papers.
Thus,
it
is
of
interest
to
pause
here
to
discuss
various
aspects
of
the
significance
of
this
diagram.
Frobenioid-theoretic
theta
monoids
Kummer
=⇒
group-theoretic
theta
monoids
Galois
Frobenioid-theoretic
Gaussian
monoids
[i.e.,
theta
values]
forget!
⇐=
⇓
evaluation
group-theoretic
Gaussian
monoids
[i.e.,
theta
values]
Fig.
3.1:
Kummer
theory
and
Galois
evaluation
(i)
The
left-hand,
central,
and
right-hand
portions
of
this
diagram
are
summa-
rized,
at
a
more
conceptual
level,
in
Fig.
3.1
above
—
that
is
to
say,
if
one
thinks
of
the
mono-theta
environments
“M
Θ
∗
”
involved
as
arising
group-theoretically
[i.e.,
from
étale-like
objects,
which
is,
of
course,
always
the
case
up
to
isomorphism!
—
cf.
the
situation
discussed
in
Corollary
3.7,
(i),
below],
then
these
portions
corre-
spond,
respectively,
to
the
arrows
“=⇒”,
“⇓”,
and
“⇐=”
in
Fig.
3.1.
Here,
we
note
that
the
final
operation
of
“forgetting”
[i.e.,
“⇐=”]
may
be
thought
of
as
the
op-
eration
of
forgetting
the
group-theoretic
—
i.e.,
“anabelian”
—
construction
of
the
Gaussian
monoids,
so
as
to
obtain
“abstract
monoids
stripped
of
any
information
concerning
the
group-theoretic
algorithms
used
to
construct
them”
—
which
we
refer
to
as
“post-anabelian”
[cf.
the
discussion
of
Remark
1.11.3,
(iii);
Corollary
3.7,
(i),
below;
the
constructions
of
Definition
3.8
below].
On
the
other
hand,
the
composite
of
the
arrows
“=⇒”
and
“⇓”
may
be
thought
of
as
a
sort
of
comparison
isomorphism
between
“Frobenius-like”
[i.e.,
“Frobenioid-
theoretic”]
and
“étale-like”
[i.e.,
“group-theoretic”]
structures
102
SHINICHI
MOCHIZUKI
—
cf.
the
discussion
of
[FrdI],
Introduction;
[IUTchI],
Corollaries
3.8,
3.9.
In
this
context,
it
is
useful
to
recall
that
the
comparison
isomorphism
of
the
“classical”
scheme-theoretic
version
of
Hodge-Arakelov
theory
[cf.
[HASurI],
Theorem
A]
is
obtained
precisely
by
evaluating
theta
functions
and
their
derivatives
at
certain
torsion
points
of
an
elliptic
curve.
(ii)
The
existence
of
both
“Frobenius-like”
and
“étale-like”
structures
in
the
theory
of
the
present
series
of
papers,
together
with
the
somewhat
complicated
theory
of
comparison
isomorphisms
as
discussed
above
in
(i),
prompts
the
following
question:
What
are
the
various
merits
and
demerits
of
“Frobenius-like”
and
“étale-
like”
structures
that
require
one
to
avail
oneself
of
both
types
of
structure
in
the
theory
of
the
present
series
of
papers
[cf.
Fig.
3.2
below]?
On
the
one
hand,
unlike
Frobenius-like
structures,
étale-like
structures
—
in
the
form
of
étale
or
tempered
fundamental
groups
[such
as
Galois
groups]
—
have
the
crucial
advantage
of
being
functorial
or
invariant
with
respect
to
various
non-
ring/scheme-theoretic
filters
between
distinct
ring/scheme
theories.
In
the
context
of
the
present
series
of
papers,
the
main
examples
of
this
phenomenon
consist
of
the
Θ-link
[cf.,
e.g.,
[IUTchI],
Corollary
3.7]
and
the
log-wall
[cf.
[Ab-
sTopIII],
§I1,
§I4;
this
theory
will
be
incorporated
into
the
present
series
of
papers
in
[IUTchIII]].
Another
important
characteristic
of
the
étale-like
structures
consti-
tuted
by
étale
or
tempered
fundamental
group
is
their
“remarkable
rigidity”
—
a
property
that
is
exhibited
explicitly
[cf.,
e.g.,
the
theory
of
[EtTh];
[AbsTopIII]]
by
various
anabelian
algorithms
that
may
be
applied
to
construct,
in
a
“purely
group-theoretic
fashion”,
various
structures
motivated
by
conventional
scheme
theory.
By
contrast,
the
Frobenius-like
structures
constituted
by
various
abstract
monoids
—
which
typically
give
rise
to
various
Frobenioids
—
satisfy
the
crucial
property
of
not
being
subject
to
such
rigidifying
anabelian
algorithms
that
re-
late
various
étale-like
structures
to
conventional
scheme
theory.
It
is
precisely
this
property
of
such
abstract
monoids
that
allows
one
to
use
these
abstract
monoids
to
construct
such
non-scheme-theoretic
filters
as
the
Θ-link
[cf.
[IUTchI],
Corollary
3.7]
or
the
log-wall
of
the
theory
of
[AbsTopIII].
Here,
it
is
interesting
to
observe
that
these
merits/demerits
of
étale-like
and
Frobenius-like
structures
play
some-
what
complementary
roles
with
respect
to
binding/not
binding
the
structures
under
consideration
to
conventional
scheme
theory.
Finally,
we
note
that
Kummer
theory
serves
the
crucial
role
[cf.
the
discussion
of
(i)]
of
relating
[via
various
comparison
isomorphisms
—
cf.
(i)]
—
within
a
given
Hodge
theater
—
potentially
non-scheme-theoretic
Frobenius-like
structures
to
étale-like
structures
which
are
subject
to
anabelian
rigidifications
that
bind
them
to
conventional
scheme
theory.
(iii)
If
one
composes
the
correspondence
“q
v
→
Θ
v
”
[cf.
the
discussion
of
[IUTchI],
Remark
3.8.1,
(i)]
constituted
by
the
Θ-link
—
i.e.,
which
relates
the
“(n
+
1)-th
generation
q-parameter”
to
the
“n-th
generation
Θ-function”
—
with
the
composite
of
the
arrows
“=⇒”,
“⇓”,
and
“⇐=”
of
Fig.
3.1,
then
one
obtains
a
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
correspondence
q
v
→
q
j
2
v
103
1≤j≤l
[cf.
Remark
2.5.1,
(i)].
In
fact,
in
the
theory
of
the
present
series
of
papers,
it
is
ultimately
this
“modified
version
of
the
Θ-link”
—
i.e.,
which
takes
into
account
the
Hodge-Arakelov-theoretic
evaluation
theory
developed
so
far
in
§2
and
the
present
§3
—
that
will
be
of
interest
to
us.
The
theory
of
this
“modified
version
of
the
Θ-link”
will
constitute
one
of
the
main
topics
treated
in
§4
below.
Here,
we
observe
that
the
above
correspondence
may
be
thought
of
as
a
sort
of
“abstract,
combinatorial
Frobenius
lifting”
—
i.e.,
as
a
sort
of
“homotopy”
between
·
the
identity
q
→
q
[i.e.,
which
corresponds
to
“characteristic
zero”]
v
v
and
·
the
purely
monoid-theoretic/highly
non-scheme-theoretic
corre-
2
spondence
q
→
q
(l
)
[i.e.,
which
corresponds
to
the
“positive
character-
v
v
istic
Frobenius
morphism”].
Moreover,
we
recall
[cf.
the
discussion
of
Remark
2.6.3]
that
the
collection
of
ex-
ponents
{j
2
}
1≤j≤l
that
appear
in
this
“abstract,
combinatorial
Frobenius
lifting”
is
highly
distinguished
—
hence,
in
particular,
far
from
arbitrary!
étale-like
structures
Frobenius-like
structures
functoriality/invariance
with
respect
to
log-wall,
Θ-link
—
rigidified
relationship
via
Kummer
theory
+
anabelian
geom.
to
conventional
arith.
geom.
—
—
lack
of
rigidification
allows
construction
of
non-scheme-theoretic
filters,
such
as
log-wall,
Θ-link
Fig.
3.2:
Étale-like
versus
Frobenius-like
structures
(iv)
In
the
context
of
the
discussion
of
(i),
it
is
of
interest
to
recall
that
vari-
ous
“Grothendieck
Conjecture-type
results”
in
anabelian
geometry
[e.g.,
over
p-adic
local
fields
and
finite
fields]
—
i.e.,
which
may
be
thought
of
as
comparison
iso-
morphisms
between
polynomial-function-theoretic
and
group-theoretic
collections
of
morphisms
—
are
obtained
precisely
by
combining
various
considerations
particular
104
SHINICHI
MOCHIZUKI
to
the
situation
of
interest
with
the
“Galois
evaluation”
via
Kummer
theory
of
polynomial
functions
or
differential
forms
at
various
rational
points
—
cf.
the
theory
of
[pGC];
[Cusp],
§2.
Remark
3.6.3.
Before
proceeding,
we
make
some
observations
concerning
base-
points
in
the
context
of
the
“non-ring/scheme-theoretic
filters”
discussed
in
Remark
3.6.2.
(i)
First,
let
us
recall
from
the
elementary
theory
of
étale
fundamental
groups
that
the
fiber
functor
associated
to
a
basepoint
is
defined
by
considering
the
points
of
a
finite
étale
covering
valued
in
some
separably
closed
field
that
lie
over
a
fixed
point
[valued
in
the
same
separably
closed
field]
of
the
base
scheme
over
which
the
covering
is
given.
Thus,
for
instance,
when
this
base
scheme
is
the
spectrum
of
a
field,
the
finite
set
of
points
associated
by
the
fiber
functor
to
a
finite
étale
covering
is
obtained
by
considering
the
various
ring
homomorphisms
from
this
field
into
some
separably
closed
field.
In
particular,
it
follows
that
the
conventional
scheme-theoretic
definition
of
a
basepoint
[in
the
form
of
a
fiber
functor]
depends,
in
an
essential
fashion,
on
the
ring/scheme
structure
of
the
rings
or
schemes
under
consideration.
One
immediate
consequence
of
these
elementary
considerations
—
which
is
of
cen-
tral
importance
in
the
theory
of
the
present
series
of
papers
—
is
the
following
ob-
servation
concerning
the
“non-ring/scheme-theoretic
filters”
discussed
in
Remark
3.6.2,
which
relate
one
ring
to
another
in
a
fashion
that
is
incompatible
with
the
respective
ring
structures:
The
distinct
ring
structures
on
either
side
of
one
of
the
“non-ring/
scheme-theoretic
filters”
discussed
in
Remark
3.6.2
—
i.e.,
the
log-wall
of
[AbsTopIII]
and
the
Θ-link
of
[IUTchI],
Corollary
3.7
—
give
rise
to
dis-
tinct,
unrelated
basepoints
[cf.
the
discussion
of
[AbsTopIII],
Remark
3.7.7,
(i)].
In
some
sense,
the
above
discussion
may
be
thought
of
as
an
“expanded,
leisurely
version”
of
an
observation
made
at
the
beginning
of
the
discussion
of
[AbsTopIII],
Remark
3.7.7,
(i).
(ii)
The
observations
of
(i)
also
apply
to
the
“N
-th
power
morphisms”
[where
N
>
1]
—
i.e.,
“morphisms
of
Frobenius
type”
—
that
appear
in
the
theory
of
Frobenioids
[cf.
[FrdI],
[FrdII],
[EtTh]].
That
is
to
say,
in
the
context
of
the
tempered
Frobenioids
that
appear
in
the
theory
of
[EtTh],
§5,
such
“morphisms
of
Frobenius
type”
[i.e.,
“N
-th
power
morphisms”
regarded
as
morphisms
contained
in
the
underlying
categories
associated
to
these
tempered
Frobenioids]
induce
“N
-th
power
morphisms”
between
various
monoids
[arising
from
the
Frobenioid
structure]
.
In
particular,
isomorphic
to
O
K
v
these
N
-th
power
morphisms
of
monoids
fail
[since
N
>
1]
to
preserve
the
ring
structure
of
K
v
,
hence
give
rise
to
distinct,
unrelated
base-
points
on
the
domain
and
codomain
objects
of
the
original
“morphism
of
Frobenius
type”
[cf.
the
discussion
of
(i)].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
105
On
the
other
hand,
let
us
observe
that
unlike
the
situations
considered
in
the
dis-
cussion
of
(i),
the
considerations
of
the
present
discussion
involving
N
-th
power
morphisms
take
place
in
a
fashion
that
is
compatible
with
the
projection
func-
tor
to
the
base
category
of
the
Frobenioid.
One
important
consequence
of
this
last
observation
is
that
unlike
the
situations
discussed
in
(i)
involving
the
log-wall
and
the
Θ-link
in
which
one
must
consider
arbitrary
isomorphisms
of
topologi-
cal
groups
between
the
étale
[or
tempered]
fundamental
groups
that
arise
in
the
domain
and
the
codomain
of
the
operation
under
consideration,
in
the
situation
of
the
present
discussion
of
N
-th
power
morphisms,
the
“distinct,
unrelated
basepoints”
that
arise
only
give
rise
to
inner
auto-
morphisms
of
the
topological
group
determined
by
[i.e.,
roughly
speak-
ing,
the
“fundamental
group”
of]
the
base
category.
This
phenomenon
may
be
thought
of
as
a
reflection
of
the
fact
that
the
application
of
an
N
-th
power
morphism
is
somewhat
“milder”
than
the
log-wall
or
Θ-link
considered
in
(i)
in
that
it
only
involves
an
operation
—
i.e.,
raising
to
the
N
-th
power
—
that
is
“algebraic”,
in
the
sense
that
it
is
defined
with
respect
to
the
ring
structure
of
the
ring
[e.g.,
K
v
]
involved.
This
somewhat
“milder
nature”
of
an
N
-th
power
morphism
allows
one
to
consider
N
-th
power
morphisms
within
a
single
category
[namely,
the
tempered
Frobenioid
under
consideration]
which
can
be
defined
in
terms
of
[formal]
flat
O
K
v
-schemes
[cf.
the
point
of
view
of
[EtTh],
§1].
By
contrast,
the
operation
inherent
in
the
log-wall
or
Θ-link
considered
in
(i)
is
much
more
drastic
and
arithmetic
[i.e.,
“non-algebraic”]
in
nature,
and
it
is
difficult
to
see
how
to
fit
such
an
operation
into
a
single
category
that
somehow
“extends”
the
tempered
Frobenioid
under
consideration
in
a
fashion
that
“lies
over”
the
same
base
category
as
the
tempered
Frobenioid
—
cf.,
e.g.,
Remark
1.11.2,
(ii),
in
the
case
of
the
Θ-link;
the
discussion
of
[AbsTopIII],
Remark
3.7.7,
in
the
case
of
the
log-wall.
Put
another
way,
the
highly
nontrivial
study
of
the
mathematical
structures
“generated
by
the
log-wall
and
Θ-link”
is,
in
some
sense,
one
of
the
main
themes
of
the
theory
of
the
present
series
of
papers
—
cf.,
especially,
the
theory
of
[IUTchIII]!
Remark
3.6.4.
Since
the
theory
of
mono-theta
environments
developed
in
[EtTh]
plays
a
fundamental
role
in
the
theory
of
the
present
paper
—
cf.,
e.g.,
Corollaries
1.12,
2.8,
3.5,
3.6
—
it
is
of
interest
to
pause
to
review
the
relationship
of
the
theory
of
[EtTh]
to
the
theory
developed
so
far
in
the
present
paper.
(i)
The
various
remarks
following
[EtTh],
Corollary
5.12,
discuss
the
signifi-
cance
of
the
various
rigidity
properties
of
a
mono-theta
environment
that
are
verified
in
[EtTh].
The
logical
starting
point
of
this
discussion
is
the
situation
considered
in
[EtTh],
Remarks
5.12.1,
5.12.2,
consisting
of
an
abstract
category
which
is
only
known
up
to
isomorphism
[i.e.,
up
to
an
indeterminate
equivalence
of
categories],
and
in
which
each
of
the
objects
is
only
known
up
to
isomorphism.
The
main
example
of
such
a
category,
in
the
context
of
the
theory
of
[EtTh],
is
a
tempered
Frobenioid
of
the
sort
considered
in
Propositions
3.3,
3.4;
Corollary
3.6.
The
situa-
tion
of
[EtTh],
Remarks
5.12.1,
5.12.2,
in
which
each
of
the
objects
in
the
category
106
SHINICHI
MOCHIZUKI
is
only
known
up
to
isomorphism,
contrasts
sharply
with
the
notion
of
a
system,
or
tower,
of
[specific!]
coverings
—
e.g.,
of
the
sort
that
appears
in
Kummer
theory,
in
which
the
coverings
are
related
by
[specific!]
N
-th
power
morphisms.
Indeed,
the
various
rigidity
properties
verified
in
[EtTh]
are
of
interest
precisely
because
they
yield
effective
reconstruction
algorithms
for
reconstructing
the
various
structures
of
interest
in
a
fashion
that
is
invariant
with
respect
to
the
indeterminacies
that
arise
from
a
situation
in
which
each
of
the
objects
in
the
category
is
only
known
up
to
isomorphism.
This
prompts
the
following
question:
What
is
the
fundamental
reason,
in
the
context
of
the
theory
of
the
present
series
of
papers,
that
one
must
work
under
the
assumption
that
each
of
the
objects
in
the
category
is
only
known
up
to
isomorphism,
thus
requiring
one
to
avail
oneself
of
the
rigidity
theory
of
[EtTh]?
To
understand
the
answer
to
this
question,
let
us
first
observe
that
Kummer
towers
involving
[specific!]
N
-th
power
morphisms
are
constructed
by
using
the
multi-
plicative
structure
of
the
“rational
functions”
[such
as
the
p
v
-adic
local
field
K
v
]
under
consideration.
That
is
to
say,
the
N
-th
power
morphisms
are
compatible
with
the
multiplicative
structure,
but
not
the
additive
structure
of
such
rational
functions.
On
the
other
hand,
ultimately,
when,
in
[IUTchIII],
we
consider
the
theory
of
the
log-wall
[cf.
[Ab-
sTopIII]],
it
will
be
of
crucial
importance
to
consider,
within
each
Hodge
theater,
the
ring
structure
[i.e.,
both
the
multiplicative
and
additive
struc-
tures]
of
the
fields
K
v
.
That
is
to
say,
without
the
ring
structure
on
K
v
,
one
cannot
even
define
the
p
v
-
adic
logarithm!
Put
another
way,
the
N
-th
power
morphisms
that
appear
in
a
Kummer
tower
may
be
thought
of
as
“Frobenius
morphisms
of
a
sort”
that
relate
distinct
ring
structures
—
i.e.,
since
the
N
-th
power
morphism
fails
to
be
compatible
with
addition!
In
particular,
the
distinct
ring
structures
that
exist
in
the
domain
and
codomain
of
such
a
“Frobenius
morphism”
necessarily
give
rise
to
distinct,
unrelated
basepoints
[cf.
the
discussion
of
Remark
3.6.3,
(ii)]
—
i.e.,
at
an
ab-
stract
category-theoretic
level,
to
objects
which
are
only
known
up
to
isomorphism!
This
is
what
requires
one
to
contend
with
the
indeterminacies
discussed
in
[EtTh],
Remarks
5.12.1,
5.12.2.
(ii)
The
theory
of
[EtTh]
may
be
summarized
as
asserting
that
one
may
re-
construct
various
structures
of
interest
from
a
mono-theta
environment
without
sacrificing
certain
fundamental
rigidity
properties,
even
in
a
situation
subject
to
certain
indeterminacies
[cf.
(i)].
Moreover,
mono-theta
environments
serve
as
a
sort
of
bridge
[cf.
[EtTh],
Remark
5.10.1]
between
tempered
Frobenioids
—
i.e.,
“Frobenius-like
structures”
[cf.
Remark
3.6.2]
—
as
in
Propositions
3.3,
3.4;
Corol-
lary
3.6,
on
the
one
hand,
and
tempered
fundamental
groups
[cf.
Proposition
3.4]
—
i.e.,
“étale-like
structures
[cf.
Remark
3.6.2]
—
on
the
other.
(iii)
One
central
feature
of
the
theory
of
[EtTh]
is
an
explanation
of
the
special
role
played
by
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function,
a
role
which
is
reflected
in
the
theory
of
cyclotomic
rigidity
developed
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
107
in
[EtTh]
[cf.
[EtTh],
Introduction].
Note
that
the
operation
of
Galois
evaluation
is
necessarily
linear
[cf.
the
discussion
of
Remark
1.12.4].
This
linearity
may
be
seen
in
the
linearity
of
the
arrows
“=⇒”,
“⇓”,
and
“⇐=”
of
Fig.
3.1.
In
particular,
these
arrows
are
compatible
with
the
ring
structure
on
the
constants
[i.e.,
“K
v
”]
—
a
property
that
will
be
of
crucial
importance
when,
in
[IUTchIII],
we
consider
the
theory
of
the
log-wall
[cf.
the
discussion
of
(i)
above].
Moreover,
this
linearity
property
of
the
operation
of
Galois
evaluation
implies
that
the
first
power
of
the
theta
values
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function
“inherits”,
so
to
speak,
the
special
role
played
by
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function.
This
observation
is
interesting
in
light
of
the
discussions
of
Remarks
2.6.3;
3.6.2,
(iii).
(iv)
In
the
context
of
(iii),
we
note
that
the
various
theta
monoids
discussed
in
Propositions
3.1,
3.3,
as
well
as
the
various
Gaussian
monoids
discussed
in
Corol-
laries
3.5,
3.6,
involve
arbitrary
powers/roots
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function.
Nevertheless,
it
is
important
to
remember
that
in
order
to
apply
the
Θ-link
—
which
requires
one
to
work
with
“Frobe-
nius-like
structures”
[cf.
the
discussion
of
Remark
3.6.2,
(ii)]
—
it
is
necessary
to
consider
the
operation
of
Galois
evaluation
summarized
in
Fig.
3.1
applied
to
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
Frobenioid-theoretic
theta
function
in
order
to
avail
oneself
of
the
cyclotomic
rigidity
furnished
by
the
delicate
bridge
constituted
by
the
mono-theta
environment
—
cf.
(ii)
above.
That
is
to
say,
the
“narrow
bridge”
afforded
by
the
mono-theta
environment
between
the
worlds
of
“Frobenius-like”
and
“étale-like”
structures
may
only
be
crossed
by
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function
and
its
theta
values.
Put
another
way,
from
the
point
of
view
of
the
étale-like
portion
[i.e.,
“group-theoretic”
portion]
of
the
operation
of
Galois
evaluation
summarized
in
Fig.
3.1,
the
N
-th
power
of
the
[reciprocal
of
the
l-th
root
of
the]
Frobenioid-theoretic
theta
function,
for
N
>
1,
is
only
defined
as
the
N
-th
power
“(−)
N
”
of
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
Frobenioid-
theoretic
theta
function.
That
is
to
say,
from
the
point
of
view
of
the
étale-like
portion
of
the
operation
of
Galois
evaluation
summarized
in
Fig.
3.1,
the
N
-th
power
of
the
[reciprocal
of
the
l-th
root
of
the]
Frobenioid-theoretic
theta
function,
for
N
>
1
—
hence,
in
particular,
the
Θ-link
—
may
only
be
calculated
by
forming
the
N
-th
power
“(−)
N
”
of
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
Frobenioid-
theoretic
theta
function.
(v)
The
necessity
of
working
with
“Frobenius-like
structures”
[cf.
the
discussion
of
(iv)]
may
also
be
thought
of
as
the
necessity
of
working
with
the
various
post-
anabelian
monoids
arising
from
the
group-theoretic
“anabelian”
algorithms
that
appear
in
the
operation
of
Galois
evaluation
[cf.
the
discussion
of
Remark
3.6.2,
108
SHINICHI
MOCHIZUKI
(i)].
In
the
context
of
this
observation,
it
is
useful
to
recall
that
from
the
point
of
view
of
the
theory
of
§1,
the
“narrow
bridge”
furnished
by
[for
instance,
the
cyclotomic
rigidity
of]
a
mono-theta
environment
satisfies
the
crucial
property
of
multira-
diality
[cf.
Corollaries
1.10,
1.12]
—
i.e.,
of
being
“horizontal”
with
respect
to
the
“connection
structure”
determined
by
the
formulation
of
this
multiradiality
[cf.
the
point
of
view
discussed
in
Remarks
1.7.1,
1.9.2].
Put
another
way,
to
work
with
powers
other
than
the
first
power
of
the
[reciprocal
of
the
l-th
root
of
the]
theta
function
or
its
theta
values
gives
rise
to
structures
which
are
“not
horizontal”
with
respect
to
this
“connection
structure”.
This
point
of
view
is
consistent
with
the
point
of
view
of
Remark
3.6.5,
(iii),
below.
A
similar
observation
concerning
multiradiality
will
also
apply
to
the
“multiradial
versions
of
the
Gaussian
monoids”
that
will
be
constructed
in
[IUTchIII]
[cf.
Remark
3.7.1
below].
Remark
3.6.5.
In
light
of
the
central
role
played
by
mono-theta-theoretic
cyclotomic
rigidity
in
the
discussion
of
Remark
3.6.4,
we
pause
to
make
some
observations
—
of
a
somewhat
more
philosophical
nature
—
concerning
this
topic.
(i)
First
of
all,
we
observe
that
a
cyclotome
may
be
thought
of
as
a
sort
of
“skeleton
of
the
arithmetic
holomorphic
structure”
under
consideration
—
cf.
the
discussion
of
Remark
1.11.6.
Indeed,
this
point
of
view
may
be
thought
of
as
being
motivated
by
the
situation
at
archimedean
primes,
where
the
circle
“S
1
”
may
be
thought
of
as
a
sort
of
“representative
skeleton
of
C
×
”.
This
point
of
view
will
play
a
central
role
in
the
remainder
of
the
discussion
of
the
present
Remark
3.6.5,
as
well
as
in
the
discussion
of
Remark
3.8.3
below.
(ii)
In
the
theory
of
[EtTh],
(a)
the
commutator
structure
[−,
−]
of
the
theta
group
plays
a
central
role
in
the
theory
of
mono-theta-theoretic
cyclotomic
rigidity
—
cf.
[EtTh],
Introduction;
[EtTh],
Remark
2.19.2.
On
the
other
hand,
in
the
classical
theory
of
algebraic
theta
functions
(b)
the
commutator
structure
[−,
−]
of
the
theta
group
plays
a
central
role
in
the
theory
via
the
observation
that
this
commutator
structure
implies
the
irreducibility
of
certain
representations
of
the
theta
group.
At
first
glance,
these
two
applications
(a),
(b)
of
the
commutator
structure
[−,
−]
of
the
theta
group
may
appear
to
be
unrelated.
In
fact,
however,
they
may
both
be
understood
as
examples
of
the
following
phenomenon:
(c)
the
commutator
structure
[−,
−]
of
the
theta
group
may
be
thought
of
as
a
sort
of
concrete
embodiment
of
the
“coherence
of
holomorphic
structures”.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
109
Indeed,
as
discussed
in
[EtTh],
Introduction,
from
the
point
of
view
of
the
scheme-
theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII],
the
irreducible
representa-
tions
that
appear
in
the
classical
theory
of
algebraic
theta
functions
as
submodules
of
the
module
of
all
set-theoretic
functions
on
the
l-torsion
points
of
an
elliptic
curve
[cf.
(b)]
may
be
thought
of,
for
instance,
when
l
is
large,
as
discrete
analogues
of
the
submodule
of
“holomorphic
functions”
within
the
module
of
all
real
analytic
functions.
On
the
other
hand,
if
one
thinks
of
cyclotomes
as
“skeleta
of
arithmetic
holomorphic
structures”
[cf.
(i)],
then
the
theory
of
conjugate
synchronization
[cf.
Remark
3.5.2,
as
well
as
Remark
3.8.3
below]
—
applied,
for
instance,
in
the
case
of
cyclotomes
—
may
be
thought
of
as
a
sort
of
“discretely
parametrized”
[in
the
sense
that
it
is
indexed
by
torsion
points]
coherence
of
arithmetic
holo-
morphic
structures,
which
is
obtained
by
working
with
the
connected
subgraph
Γ
X
⊆
Γ
X
[cf.
Remark
2.6.1,
(i)].
In
this
context,
mono-theta-theoretic
cyclotomic
rigidity
[cf.
(a)]
may
be
thought
of
as
a
sort
of
“continuously
parametrized
version”
[i.e.,
supported
on
Ÿ
v
,
as
opposed
to
a
finite
set
of
torsion
points]
of
this
coherence
of
arithmetic
holomorphic
structures.
Finally,
we
recall
that
the
interaction
—
i.e.,
via
restriction
operations
—
between
these
“discrete”
and
“continuous”
versions
of
the
“coherence
of
arithmetic
holomorphic
structures”
plays
a
central
role
in
the
theory
of
Galois
evaluation
given
in
Corollaries
2.8,
(i);
3.5,
(ii);
3.6,
(ii).
(iii)
If
one
thinks
of
cyclotomes
at
localizations
[say,
at
v
∈
V
bad
]
of
a
number
field
[i.e.,
K]
as
local
skeleta
of
the
arithmetic
holomorphic
structure
[cf.
(i)],
then
the
mono-theta-theoretic
cyclotomic
rigidity
may
be
thought
of
as
a
sort
of
“local
uniformization”
of
a
number
field
[cf.
the
exterior
cyclo-
tome
of
a
mono-theta
environment
that
arises
from
a
tempered
Frobenioid,
as
in
Proposition
1.3,
(i)]
via
a
local
portion
[cf.
the
interior
cyclotome
in
the
situation
of
Proposition
1.3,
(i)]
of
the
geometric
tempered
funda-
mental
group
Δ
v
associated
to
a
certain
covering
of
the
once-punctured
elliptic
curve
X
F
[cf.
Definition
2.3,
(i);
[IUTchI],
Definition
3.1,
(e)].
Since
the
cyclotomic
rigidity
isomorphism
arising
from
mono-theta-theoretic
cyclo-
tomic
rigidity
may
be
thought
of
as
the
“cyclotomic
portion”
of
the
theta
function,
mono-theta-theoretic
cyclotomic
rigidity
may
be
interpreted
as
the
statement
that
the
theta
function
constructed
from
a
mono-theta
environment
is
free
of
any
Z
×
-
power
indeterminacies.
Moreover,
if
one
takes
this
point
of
view,
then
constant
multiple
rigidity
may
be
thought
of
as
the
statement
that
the
above
“local
uniformization”
is
sufficiently
rigid
as
to
be
free
of
any
constant
multiple
indeterminacies.
Here,
it
is
useful
to
recall
that
the
once-punctured
elliptic
curve
X
F
on
the
number
field
F
that
occurs
in
the
theory
of
the
present
series
of
papers
may
be
thought
of
as
being
analogous
to
the
nilpotent
ordinary
indigenous
bundles
on
a
hyperbolic
curve
in
positive
characteristic
in
p-adic
Teichmüller
theory
[cf.
the
discussion
of
[AbsTopIII],
§I5].
That
it
to
say,
from
this
point
of
view,
the
“local
uniformiza-
tions”
of
the
above
discussion
may
be
thought
of
as
corresponding
to
the
local
uniformizations
via
canonical
coordinates
of
p-adic
Teichmüller
theory
[cf.,
e.g.,
[pTeich],
§0.9],
which
are
also
“sufficiently
rigid”
as
to
be
free
of
any
Z
×
-power
or
constant
multiple
indeterminacies.
Here,
mono-theta-theoretic
cyclotomic
rigid-
ity
may
be
thought
of
as
corresponding
to
the
Kodaira-Spencer
isomorphism
110
SHINICHI
MOCHIZUKI
[associated
to
the
Hodge
section
of
the
canonical
indigenous
bundle],
which,
in
some
sense,
may
be
thought
of
as
the
“skeleton”
of
the
local
uniformizations
of
p-adic
Teichmüller
theory.
Also,
it
is
useful
to
recall
in
this
context
that
the
canonical
coordinates
of
p-adic
Teichmüller
theory
are
constructed
by
considering
invariants
with
respect
to
certain
canonical
Frobenius
liftings.
Put
another
way,
the
technique
of
considering
Frobenius-invariants
allows
one
to
pass,
in
a
canonical
way,
from
objects
defined
modulo
p
to
objects
defined
modulo
higher
powers
of
p.
Since
the
various
Θ-links
of
the
Frobenius-picture
may
be
regarded
as
corresponding
to
the
various
transitions
from
“mod
p
n
to
mod
p
n+1
”
[where
n
∈
N]
in
the
theory
of
Witt
vectors
[cf.
the
discussion
of
[IUTchI],
§I4;
[IUTchIII],
Remark
1.4.1,
(iii)],
it
is
natural
to
regard,
in
the
context
of
the
canonical
splittings
furnished
by
the
étale-picture
[cf.
the
discussion
of
[IUTchI],
§I1],
the
multiradiality
of
the
formulation
of
mono-theta-theoretic
cyclotomic
rigidity
and
constant
multiple
rigidity
given
in
Corollary
1.12
as
corre-
sponding
to
the
Frobenius-invariant
nature
of
the
canonical
coordinates
of
p-adic
Teichmüller
theory.
Finally,
in
this
context,
we
observe
that
it
is
perhaps
natural
to
think
of
the
dis-
crete
rigidity
of
the
theory
of
[EtTh]
as
corresponding
to
the
fact
that
the
canoni-
cal
coordinates
of
p-adic
Teichmüller
theory,
which
a
priori
may
only
be
constructed
as
PD-formal
power
series,
may
in
fact
be
constructed
as
power
series
in
the
usual
sense,
i.e.,
elements
of
the
completion
O
of
the
local
ring
at
the
point
under
consideration.
Indeed,
the
discrete
rigidity
of
[EtTh]
implies
that
one
may
restrict
oneself
to
working
with
the
usual
theta
function,
canonical
multiplicative
coordi-
nates
[i.e.,
“U
”],
and
q-parameters
on
appropriate
tempered
coverings
of
the
Tate
curve,
all
of
which,
like
the
power
series
arising
from
canonical
parameters
in
p-adic
Teichmüller
theory,
give
rise
to
“functions
on
suitable
formal
schemes”
in
the
sense
of
classical
scheme
theory.
By
contrast,
if
this
discrete
rigidity
were
to
fail,
then
one
would
be
obliged
to
work
in
an
“a
priori
profinite”
framework
that
involves,
for
in-
stance,
Z-powers
of
“U
”
and
“q”
[cf.
[EtTh],
Remarks
1.6.4,
2.19.4].
Such
Z-powers
appear
naturally
in
the
Z-modules
that
arise
[e.g.,
as
cohomology
modules]
in
the
Kummer
theory
of
the
theta
function
and
may
be
thought
of
as
corresponding
to
PD-formal
power
series
in
the
sense
that
arbitrary
O-powers
of
canonical
parame-
ters
[say,
for
simplicity,
at
non-cuspidal
ordinary
points
of
a
canonical
curve],
which
arise
naturally
when
one
considers
such
parameters
additively
[cf.
the
discussion
of
“canonical
affine
coordinates”
in
[pOrd],
Chapter
III],
cannot
be
defined
if
one
restricts
oneself
to
working
with
conventional
power
series
—
i.e.,
such
O-powers
may
only
be
defined
if
one
allows
oneself
to
work
with
PD-formal
power
series.
Corollary
3.7.
(Group-theoretic
Gaussian
Monoids
and
Uniradiality)
Suppose
that
we
are
in
the
situation
of
Proposition
3.4,
i.e.,
in
the
following,
we
consider
the
full
poly-isomorphism
M
Θ
∗
(Π
v
)
∼
→
†
M
Θ
∗
(
F
v
)
—
where
M
Θ
∗
(Π
v
)
is
the
projective
system
of
mono-theta
environments
arising
from
the
algorithm
of
Proposition
1.2,
(i)
[cf.
also
Proposition
1.5,
(i)];
†
F
v
is
a
tem-
pered
Frobenioid
as
in
Proposition
3.3
—
of
projective
systems
of
mono-
Θ
†
theta
environments.
When
“M
Θ
∗
”
is
taken
to
be
M
∗
(
F
v
),
we
shall
denote
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
111
Θ
†
Θ
resulting
“M
Θ
¨
”
by
M
∗
¨
(
F
v
)
[cf.
Definition
2.7,
(ii)].
When
“M
∗
”
is
taken
to
∗
Θ
Θ
be
M
Θ
¨
(M
∗
∗
(Π
v
),
we
shall
identify
Π
v
¨
)
and
G
v
(M
∗
¨
)
[cf.
Definition
2.7,
(ii)]
with
Π
v
¨
and
G
v
(Π
v
¨
)
[cf.
Corollary
2.5,
(i)],
respectively,
via
the
tautological
∼
∼
Θ
Θ
isomorphisms
Π
v
¨
(M
∗
¨
,
G
v
(M
∗
¨
).
Finally,
we
shall
follow
¨
)
→
Π
v
¨
)
→
G
v
(Π
v
the
notational
conventions
of
Corollaries
3.5,
3.6
with
regard
to
the
subscripts
“|t|”,
for
|t|
∈
|F
l
|,
and
“F
l
”.
(i)
(From
Group-theoretic
to
Post-anabelian
Gaussian
Monoids)
Each
∼
Θ
†
isomorphism
of
projective
systems
of
mono-theta
environments
M
Θ
∗
(Π
v
)
→
M
∗
(
F
v
)
induces
compatible
[in
the
evident
sense]
collections
of
isomorphisms
Π
v
¨
{G
v
(Π
v
¨
)
|t|
}
|t|∈F
l
∼
ι
Θ
∞
Ψ
env
(M
∗
(Π
v
))
→
Ψ
ι
env
(M
Θ
∗
(Π
v
))
→
∼
∼
→
Θ
∞
Ψ
ξ
(M
∗
(Π
v
))
Ψ
ξ
(M
Θ
∗
(Π
v
))
∼
†
{G
v
(M
Θ
¨
(
F
v
))
|t|
}
|t|∈F
∗
†
{G
v
(M
Θ
∗
(
F
v
))
|t|
}
|t|∈F
→
l
l
∼
∼
→
Θ
†
∞
Ψ
ξ
(M
∗
(
F
v
))
→
†
∞
Ψ
F
ξ
(
F
v
)
∼
†
Ψ
ξ
(M
Θ
∗
(
F
v
))
→
∼
Ψ
F
ξ
(
†
F
v
)
→
and
G
v
(Π
v
¨
)
∼
→
×
Ψ
ι
env
(M
Θ
∗
(Π
v
))
G
v
(Π
v
¨
)
F
l
∼
→
×
Ψ
ξ
(M
Θ
∗
(Π
v
))
∼
→
†
G
v
(M
Θ
¨
(
F
v
))
F
∗
l
∼
→
∼
→
†
×
Ψ
ξ
(M
Θ
∗
(
F
v
))
†
G
v
(M
Θ
∗
(
F
v
))
F
l
∼
→
Ψ
F
ξ
(
†
F
v
)
×
—
where
the
upper
left-hand
portion
of
the
first
display
[involving
“”]
is
obtained
by
applying
the
third
display
[involving
“”]
of
Corollary
3.5,
(ii),
in
Θ
the
case
where
“M
Θ
∗
”
is
taken
to
be
M
∗
(Π
v
);
the
isomorphisms
that
relate
the
upper
left-hand
portion
of
the
first
display
to
the
lower
right-hand
portion
of
the
first
display
arise
from
the
functoriality
of
the
algorithms
involved,
relative
to
isomorphisms
of
projective
systems
of
mono-theta
environments;
the
lower
right-
hand
portion
of
the
first
display
is
obtained
by
applying
the
right-hand
portion
112
SHINICHI
MOCHIZUKI
of
the
third
display
of
Corollary
3.6,
(ii),
in
the
case
where
“M
Θ
∗
”
is
taken
to
be
Θ
†
M
∗
(
F
v
);
the
second
display
is
obtained
from
the
first
display
by
considering
the
units
[denoted
by
means
of
a
superscript
“×”].
(ii)
(Uniradiality
of
Gaussian
Monoids)
If
we
write
Ψ
F
ξ
(
†
F
v
)
×μ
for
the
ind-topological
monoid
obtained
by
forming
the
quotient
of
Ψ
F
ξ
(
†
F
v
)
×
by
its
torsion
subgroup,
then
the
functorial
algorithms
Π
v
→
Ψ
gau
(M
Θ
∗
(Π
v
));
Π
v
→
∞
Ψ
gau
(M
Θ
∗
(Π
v
))
Θ
—
where
we
think
of
Ψ
gau
(M
Θ
∗
(Π
v
)),
∞
Ψ
gau
(M
∗
(Π
v
))
as
being
equipped
with
their
natural
splittings
up
to
torsion
[cf.
Corollary
3.5,
(iii)]
and,
in
the
case
of
Ψ
gau
(M
Θ
¨
)-action
[cf.
Corollary
3.5,
(ii)]
—
obtained
by
∗
(Π
v
)),
the
natural
G
v
(Π
v
composing
the
algorithms
of
Proposition
1.2,
(i);
Corollary
3.5,
(ii),
(iii),
depend
on
the
cyclotomic
rigidity
isomorphism
of
Corollary
1.11,
(b)
[cf.
Remark
1.11.5,
(ii);
the
use
of
the
surjection
of
Remark
1.11.5,
(i),
in
the
algorithms
of
Proposition
3.1,
(ii),
and
Corollary
3.5,
(ii)],
hence
fail
to
be
compatible,
rela-
tive
to
the
displayed
diagrams
of
(i),
with
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-topological
monoid
equipped
with
the
action
of
a
topological
group,
determined
by]
the
pair
†
G
v
(M
Θ
∗
(
F
v
))
F
l
Ψ
F
ξ
(
†
F
v
)
×μ
which
arise
from
automorphisms
of
[the
underlying
pair,
consisting
of
an
ind-
topological
monoid
equipped
with
the
action
of
a
topological
group,
determined
by]
†
†
×
the
pair
G
v
(M
Θ
∗
(
F
v
))
F
Ψ
F
ξ
(
F
v
)
[cf.
Remarks
1.11.1,
(i),
(b);
1.8.1]
—
l
in
the
sense
that
this
algorithm,
as
given,
only
admits
a
uniradial
formulation
[cf.
Remarks
1.11.3,
(iv);
1.11.5,
(ii)].
Proof.
The
various
assertions
of
Corollary
3.7
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
3.7.1.
One
central
consequence
of
the
theory
to
be
developed
in
[IUTchIII]
[cf.
Remarks
2.9.1,
(iii);
3.4.1,
(ii)]
is
the
result
that,
by
applying
the
theory
of
log-shells
[cf.
[AbsTopIII]],
one
may
modify
the
algorithms
of
Corollary
3.7,
(ii),
in
such
a
way
as
to
obtain
algorithms
for
computing
the
Gaussian
monoids
that
[yield
functors
which]
are
manifestly
multiradially
defined
—
albeit
at
the
cost
of
allowing
for
certain
[relatively
mild!]
indeterminacies.
The
following
definition
in
some
sense
summarizes
the
theory
of
the
present
§3.
Definition
3.8.
Many
of
the
“monoids
equipped
with
a
Galois
action”
that
appear
in
the
discussion
of
the
present
§3
may
be
thought
of
as
giving
rise
to
Frobenioids,
as
follows.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
113
(i)
Each
of
the
monoids
equipped
with
a
Π
X
(M
Θ
∗
)-action
Π
X
(M
Θ
∗
)
Ψ
cns
(M
Θ
∗
);
Π
X
(M
Θ
∗
)
Ψ
†
C
v
of
Propositions
3.1,
(ii);
3.3,
(ii),
gives
rise
to
a
p
v
-adic
Frobenioid
of
monoid
type
Z
[cf.
[FrdII],
Example
1.1,
(ii)]
F
†
C
v
F
cns
(M
Θ
∗
);
0
whose
divisor
monoid
associates
to
every
object
of
B
temp
(Π
X
(M
Θ
∗
))
a
monoid
isomorphic
to
Q
≥0
.
It
follows
immediately
from
the
construction
of
the
data
“Π
X
(M
Θ
∗
)
Ψ
†
C
v
”
[cf.
Example
3.2,
(ii)]
that
one
has
a
tautological
isomor-
phism
of
Frobenioids
∼
†
C
v
→
F
†
C
v
[cf.
the
discussion
of
[IUTchI],
Example
3.2,
(iii),
(iv)],
which
we
shall
use
to
identify
these
two
Frobenioids.
Thus,
the
isomorphism
of
monoids
of
Proposition
3.3,
(ii),
may
be
interpreted
as
an
isomorphism
of
Frobenioids
†
C
v
∼
→
F
cns
(M
Θ
∗
)
∼
—
which
also
admits
[indeed,
induces]
a
“mono-analytic
version”
†
C
v
→
F
cns
(M
Θ
∗
)
[cf.
the
category
“C
v
”
of
[IUTchI],
Example
3.2,
(iv)].
This
mono-analytic
version
admits
a
“labeled
version”
[cf.
Remark
3.8.1
below]
(
†
C
v
)
|t|
∼
→
(F
cns
(M
Θ
∗
))
|t|
—
cf.
Corollary
3.6,
(i).
Finally,
one
has
Frobenioid-theoretic
interpretations
(F
cns
(M
Θ
∗
))
|F
l
|
;
∼
(F
cns
(M
Θ
∗
))
0
(
†
C
v
)
|F
l
|
;
(
†
C
v
)
0
(F
cns
(M
Θ
∗
))
F
→
∼
→
l
(
†
C
v
)
F
l
of
the
constructions
of
Corollary
3.5,
(iii);
3.6,
(iii).
(ii)
Each
of
the
monoids
equipped
with
a
topological
group
action
ι
Θ
G
v
(M
Θ
¨
)
Ψ
env
(M
∗
);
∗
Θ
G
v
(M
Θ
¨
)
F
Ψ
ξ
(M
∗
);
∗
l
G
v
(M
Θ
¨
)
Ψ
†
F
v
Θ
,α
∗
†
G
v
(M
Θ
∗
)
F
Ψ
F
ξ
(
F
v
)
l
[cf.
Proposition
3.1,
(i);
Proposition
3.3,
(i);
Corollary
3.5,
(ii);
Corollary
3.6,
(ii)]
gives
rise
to
a
p
v
-adic
Frobenioid
of
monoid
type
Z
[cf.
[FrdII],
Example
1.1,
(ii)]
ι
(M
Θ
F
env
∗
);
F
†
F
v
Θ
,α
;
F
ξ
(M
Θ
∗
);
F
F
ξ
(
†
F
v
)
whose
divisor
monoid
associates
to
every
object
of
B
temp
(G
v
(−))
0
[where
“(−)”
is
Θ
M
Θ
¨
or
M
∗
]
a
monoid
isomorphic
to
N.
Moreover,
each
of
these
Frobenioids
is
∗
114
SHINICHI
MOCHIZUKI
equipped
with
a
collection
of
splittings
[cf.
Proposition
3.1,
(i);
Proposition
3.3,
(i);
Corollary
3.5,
(iii);
Corollary
3.6,
(iii)].
Also,
we
shall
write
def
def
Θ
ι
Θ
F
env
(M
∗
)
;
F
†
F
v
Θ
=
F
†
F
v
Θ
,α
F
env
(M
∗
)
=
ι
α
def
Θ
def
Θ
†
†
F
gau
(M
∗
)
=
F
ξ
(M
∗
)
;
F
F
gau
(
F
v
)
=
F
F
ξ
(
F
v
)
ξ
ξ
[cf.
the
notation
of
Proposition
3.1,
(i);
Proposition
3.3,
(i);
Corollary
3.5,
(ii);
Corollary
3.6,
(ii)].
It
follows
immediately
from
the
construction
of
the
data
“G
v
(M
Θ
¨
)
Ψ
†
F
v
Θ
,α
”
[cf.
Example
3.2,
(i)]
that
one
has
a
tautological
iso-
∗
morphism
of
Frobenioids
∼
†
Θ
C
v
→
F
†
F
v
Θ
,α
which
is
compatible
with
the
associated
splittings
[cf.
the
discussion
of
[IUTchI],
Ex-
ample
3.2,
(v)],
and
which
we
shall
use
to
identify
these
two
split
Frobenioids.
Thus,
the
isomorphisms
of
monoids
in
the
bottom
line
of
the
third
display
of
Corollary
3.6,
(ii),
may
be
interpreted
as
isomorphisms
of
split
Frobenioids
∼
∼
ι
F
env
(M
Θ
∗
)
→
F
†
F
v
Θ
,α
→
∼
F
ξ
(M
Θ
∗
)
→
F
F
ξ
(
†
F
v
)
[cf.
Proposition
3.3,
(i);
Corollary
3.5,
(iii);
Corollary
3.6,
(iii)]
which
are
compatible
with
the
subcategories
F
2l·ξ
(M
Θ
∗
)
⊆
F
F
2l·ξ
(
†
F
v
)
F
ξ
(M
Θ
∗
);
⊆
F
F
ξ
(
†
F
v
)
determined
by
the
submonoids
“Ψ
2l·ξ
(−)”
[cf.
Corollaries
3.5,
(ii);
3.6,
(ii)]
and
which
yield
isomorphisms
of
collections
of
split
Frobenioids
F
†
F
v
Θ
∼
→
F
env
(M
Θ
∗
)
∼
→
F
gau
(M
Θ
∗
)
∼
F
F
gau
(
†
F
v
)
→
[cf.
the
fourth
display
of
Corollary
3.6,
(ii)].
†
(iii)
The
direct
products
in
which
the
submonoids
Ψ
ξ
(M
Θ
∗
)
and
Ψ
F
ξ
(
F
v
)
are
constructed
[cf.
the
second
display
of
Corollary
3.5,
(ii);
the
first
display
of
Corollary
3.6,
(ii)]
determine
natural
embeddings
of
categories
[cf.
Remark
3.8.1
below]
F
ξ
(M
Θ
∗
)
→
F
cns
(M
Θ
∗
)
|t|
;
F
F
ξ
(
†
F
v
)
|t|∈F
l
→
(
†
C
v
)
|t|
|t|∈F
l
Θ
†
†
which
coincide
on
the
subcategories
F
2l·ξ
(M
Θ
∗
)
⊆
F
ξ
(M
∗
),
F
F
2l·ξ
(
F
v
)
⊆
F
F
ξ
(
F
v
).
We
shall
write
[cf.
Remark
3.8.1
below]
F
gau
(M
Θ
∗
)
→
F
F
gau
(
†
F
v
)
F
cns
(M
Θ
∗
)
F
=
def
l
→
F
cns
(M
Θ
∗
)
|t|
|t|∈F
l
(
†
C
v
)
F
=
def
l
|t|∈F
l
(
†
C
v
)
|t|
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
115
for
the
collections
of
embeddings
of
categories
obtained
by
allowing
ξ
to
vary.
These
embeddings
may
be
thought
of
as
“Gaussian
distributions”
and
are
depicted
in
Fig.
3.3
below.
In
this
context,
it
is
useful
to
observe
that
we
also
have
natural
diagonal
embeddings
of
categories,
i.e.,
“constant
distributions”
[cf.
Remark
3.8.1
below]
∼
Θ
(M
Θ
F
cns
∗
)
→
F
cns
(M
∗
)
F
l
†
C
v
∼
→
(
†
C
v
)
F
l
→
→
F
cns
(M
Θ
∗
)
F
=
l
|t|∈F
l
(
†
C
v
)
F
=
l
F
cns
(M
Θ
∗
)
|t|
(
†
C
v
)
|t|
|t|∈F
l
∼
—
where
the
“
→
’s”
denote
the
tautological
isomorphisms
—
cf.
the
discussion
[and
notational
conventions!]
of
[IUTchI],
Example
5.4,
(i);
[IUTchI],
Fig.
5.1.
◦
◦
n
·
◦
...
◦
◦
...
◦
...
◦
·
v
..
.
◦
◦
n
·
◦
...
◦
◦
...
◦
...
◦
·
v
..
.
◦
◦
n
·
◦
...
◦
◦
...
◦
...
◦
·
v
Fig.
3.3:
Gaussian
distribution
Remark
3.8.1.
In
the
present
series
of
papers,
we
follow
the
convention
[cf.
[IUTchI],
§0]
that
an
“isomorphism
of
categories”
is
to
be
understood
as
an
isomor-
phism
class
of
equivalences
of
categories.
On
the
other
hand,
in
the
context
of
the
discussion
of
Frobenioids
in
Definition
3.8,
in
order
to
obtain
a
precise
“Frobenioid-
theoretic
translation”
of
the
results
obtained
so
far
[in
the
language
of
monoids]
that
involve
the
phenomenon
of
conjugate
synchronization
[cf.
Remark
3.5.2;
the
discussion
of
Remark
3.8.3
below],
one
is
obliged
to
consider
the
various
Frobenioids
indexed
by
a
subscript
“|t|
∈
|F
l
|”
as
being
determined
up
to
an
isomorphism
of
the
identity
functor
—
i.e.,
corresponding
to
an
“inner
automorphism”
in
the
context
of
Corollaries
3.5,
(i);
3.6,
(i)
—
which
is
independent
of
|t|
∈
|F
l
|.
In
particular,
when
there
is
a
danger
of
confusion,
perhaps
the
simplest
approach
is
to
resort
to
the
original
“monoid-theoretic
formulations”
of
Corollaries
3.5,
3.6.
116
SHINICHI
MOCHIZUKI
Remark
3.8.2.
At
this
point,
it
is
of
interest
to
pause
to
discuss
the
relation-
ship
between
the
theory
of
the
present
§3
and
the
theories
of
F
±
l
-symmetry
[cf.
[IUTchI],
§6]
and
F
l
-symmetry
[cf.
[IUTchI],
§4,
§5]
developed
in
[IUTchI].
(i)
First
of
all,
the
construction
algorithms
for
the
Gaussian
monoids
dis-
cussed
in
Corollaries
3.5,
(ii);
3.6,
(ii),
as
well
as
for
the
closely
relating
splittings
discussed
in
Corollaries
3.5,
(iii);
3.6,
(iii),
involve
restriction
to
the
decompo-
sition
groups
of
torsion
points
indexed
[via
a
functorial
algorithm]
by
profinite
conjugacy
classes
of
cusps
[cf.
Corollary
2.4,
(ii)]
which
are
subject
to
a
certain
±
F
±
l
-symmetry
[cf.
Corollary
2.4,
(iii)].
This
F
l
-symmetry
may
be
thought
of
as
the
restriction,
to
the
portion
labeled
by
the
valuation
v
∈
V
bad
under
consid-
eration,
of
the
F
±
l
-symmetry
[cf.
[IUTchI],
Proposition
6.8,
(i)]
associated
to
a
±ell
D-Θ
-Hodge
theater
[cf.
Remark
2.6.2,
(i)].
From
the
point
of
view
of
the
issue
of
“which
portion
of
the
original
once-punctured
elliptic
curve
over
a
number
field
X
F
[cf.
[IUTchI],
Definition
3.1]
is
involved”,
this
theory
of
split
Gaussian
monoids
revolves
around
various
labeled
[i.e.,
by
elements
of
copies
of
F
l
or
|F
l
|]
copies
of
the
local
Frobenioids
at
v
of
the
mono-analyticizations
of
the
F-prime-strips
that
appear
in
a
D-Θ
±ell
-Hodge
theater
—
cf.
the
various
natural
embeddings
dis-
cussed
in
Definition
3.8,
(iii)
—
i.e.,
more
concretely,
copies
of
the
portion
of
the
pair
“G
v
(Π
v
)
O
F
”
determined
by
a
certain
submonoid
of
O
F
.
Finally,
we
recall
v
v
that
after
one
executes
these
construction
algorithms
for
split
Gaussian
monoids
and
observes
the
F
±
l
-symmetry
discussed
above,
one
may
then
form
holomorphic
or
mono-analytic
processions,
indexed
by
subsets
of
|F
l
|,
as
discussed
in
[IUTchI],
Proposition
6.9,
(i),
(ii).
(ii)
On
the
other
hand,
by
applying
the
algorithm
of
[IUTchI],
Proposition
6.7,
one
may
pass
to
the
local
portion
at
v
∈
V
bad
of
a
D-ΘNF-Hodge
theater.
At
the
level
of
labels,
this
amounts
to
removing
the
label
0
∈
|F
l
|
and
identifying
this
label
with
the
complement
of
0
in
|F
l
|,
i.e.,
with
F
l
—
cf.
the
assignment
“
0,
→
>
”
of
D-prime-strips
discussed
in
[IUTchI],
Proposition
6.7.
At
the
level
of
local
Frobe-
nioids
at
v
∈
V
bad
[i.e.,
copies
of
the
pair
“Π
v
O
F
”]
corresponding
to
these
v
labels,
this
assignment
may
be
thought
of
as
corresponding
to
the
isomorphisms
∼
∼
Θ
of
monoids
“Ψ
cns
(M
Θ
∗
)
0
→
Ψ
cns
(M
∗
)
F
”
and
“(Ψ
†
C
v
)
0
→
(Ψ
†
C
v
)
F
”
dis-
l
l
cussed
in
the
first
displays
of
Corollaries
3.5,
(iii);
3.6,
(iii).
This
newly
obtained
situation
involving
the
local
portion
at
v
∈
V
bad
of
a
D-ΘNF-Hodge
theater
admits
an
F
l
-symmetry
[cf.
[IUTchI],
Proposition
4.9,
(i)]
—
cf.
the
discussion
of
the
±
F
l
-symmetry
in
the
situation
of
(i).
As
we
shall
see
in
§4
below,
at
least
at
the
level
of
value
groups,
this
newly
obtained
situation
involving
F
l
-symmetries
is
bad
to
the
valuations
well-suited
to
relating
the
theory
of
the
present
§3
at
v
∈
V
∈
V
good
,
as
well
as
to
the
global
theory
of
[IUTchI],
§5.
This
global
theory
satisfies
the
crucial
property
that
it
allows
one
to
relate
the
multiplicative
and
additive
structures
of
a
global
number
field
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.2;
[IUTchI],
Remark
6.12.5,
(ii)].
Finally,
starting
from
this
newly
obtained
situation,
one
may
proceed
to
form
holomorphic
or
mono-analytic
processions,
indexed
by
subsets
of
F
l
,
as
discussed
in
[IUTchI],
Proposition
4.11,
(i),
(ii),
which
are
com-
patible
[cf.
[IUTchI],
Proposition
6.9,
(iii)]
with
the
“|F
l
|-processions”
discussed
in
(i).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
117
Remark
3.8.3.
One
central
theme
of
the
theory
of
the
present
§3
is
the
ap-
plication
of
the
phenomenon
of
conjugate
synchronization
[cf.
Remark
3.5.2],
which
plays
a
fundamental
role
in
the
theory
of
the
group-theoretic
version
of
Hodge-Arakelov-theoretic
evaluation
of
the
theta
function
developed
in
§2.
Thus,
it
is
of
interest
to
pause
to
discuss
precisely
what
was
gained
in
the
present
§3
by
applying
the
conjugate
synchronization
obtained
in
§2.
(i)
We
begin
our
discussion
by
reviewing
the
following
direct
technical
conse-
quences
of
the
conjugate
synchronization
discussed
in
Remark
3.5.2:
(a)
the
isomorphisms
of
monoids
∼
Θ
Ψ
cns
(M
Θ
∗
)
|t
1
|
→
Ψ
cns
(M
∗
)
|t
2
|
;
∼
(Ψ
†
C
v
)
|t
1
|
→
(Ψ
†
C
v
)
|t
2
|
;
∼
(Ψ
†
C
v
)
|t|
→
Ψ
cns
(M
Θ
∗
)
|t|
—
where
|t|,
|t
1
|,
|t
2
|
∈
|F
l
|;
the
third
isomorphism
is
well-defined
up
to
an
inner
automorphism
indeterminacy
that
is
independent
of
|t|
—
dis-
cussed
in
Corollaries
3.5,
(i);
3.6,
(i);
(b)
the
construction
of
well-defined
diagonal
submonoids
Ψ
cns
(M
Θ
Ψ
cns
(M
Θ
Ψ
cns
(M
Θ
∗
)
|F
l
|
⊆
∗
)
|t|
;
∗
)
F
⊆
l
|t|∈|F
l
|
Ψ
cns
(M
Θ
∗
)
|t|
|t|∈F
l
in
Corollary
3.5,
(i),
and
the
corresponding
diagonal
embeddings
of
cate-
gories
—
i.e.,
“constant
distributions”
—
discussed
in
Definition
3.8,
(iii);
(c)
the
well-defined
isomorphisms
of
monoids
∼
Θ
Ψ
cns
(M
Θ
∗
)
0
→
Ψ
cns
(M
∗
)
F
;
l
∼
(Ψ
†
C
v
)
0
→
(Ψ
†
C
v
)
F
l
of
Corollaries
3.5,
(iii);
3.6,
(iii);
(d)
the
restriction
to
the
units
of
the
[composite]
isomorphism
of
monoids
∼
Ψ
†
F
v
Θ
,α
→
Ψ
F
ξ
(
†
F
v
)
that
appears
in
the
third
display
of
Corollary
3.6,
(ii)
[cf.
also
Fig.
3.1;
the
discussion
of
Remark
3.6.2,
(i)].
Here,
we
observe
that
(b)
and
(c)
may
be
thought
of
as
formal
consequences
of
(a),
while
(d)
may
be
thought
of
as
an
alternate
formulation
of
the
portion
of
(a)
concerning
the
units
in
the
case
of
|t|
∈
F
l
.
Moreover,
as
discussed
in
Remark
3.6.2,
(iii),
ultimately,
in
the
present
series
of
papers,
we
shall
be
interested
in
composing
the
Θ-link
with
the
composite
of
the
arrows
“=⇒”,
“⇓”,
and
“⇐=”
of
Fig.
3.1
—
i.e.,
with
the
isomorphism
of
monoids
that
appears
in
the
display
of
(d).
Indeed,
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
our
main
application
[cf.
§4
below]
of
the
conjugate
synchronization
discussed
in
Remark
3.5.2
will
consist
precisely
of
the
isomorphism
of
units
of
(d),
in
the
context
of
composition
with
the
Θ-link
—
cf.
the
“coricity
of
O
×
”
given
in
[IUTchI],
Corollary
3.7,
(iii).
118
SHINICHI
MOCHIZUKI
Finally,
in
this
context,
we
recall
that
the
isomorphisms
of
monoids
that
appear
in
the
Θ-link
or
in
the
third
display
of
Corollary
3.6,
(ii),
only
make
sense
if
one
works
with
post-anabelian
abstract
monoids/Frobenioids
—
i.e.,
with
“Frobenius-like”
structures
[cf.
the
discussion
of
Remark
3.6.2,
(i),
(ii)].
(ii)
In
[IUTchIII],
it
will
be
of
central
importance
to
consider
the
theory
of
the
present
paper
in
the
context
of
the
log-wall
[i.e.,
the
situation
considered
in
[AbsTopIII]].
In
the
context
of
the
log-wall,
it
will
be
of
fundamental
importance
to
construct
versions
of
the
various
Frobenioid-theoretic
theta
and
Gaussian
monoids
that
appeared
in
the
discussion
at
the
end
of
(i)
that
are
capable
of
“penetrating
the
log-wall”
[cf.
the
discussion
of
[AbsTopIII],
§I4]
—
i.e.,
to
construct
étale-like
ver-
sions
of
these
Frobenioid-theoretic
theta
and
Gaussian
monoids,
by
availing
oneself
of
the
right-hand
portion
of
Fig.
3.1.
Now
to
pass
from
these
Frobenioid-theoretic
monoids
to
their
étale-like
counterparts,
one
must
apply
Kummer
theory
—
cf.
the
arrow
“=⇒”
of
Fig.
3.1.
Moreover,
in
order
to
apply
Kummer
theory,
one
must
avail
oneself
of
the
cyclotomes
contained
in
[i.e.,
the
torsion
subgroups
of]
the
various
groups
of
units
of
the
relevant
monoids.
It
is
at
this
point
that
it
is
necessary
to
apply,
in
the
fashion
discussed
in
(i),
the
conjugate
synchroniza-
tion
discussed
in
Remark
3.5.2
in
an
essential
way.
That
is
to
say,
if
one
is
in
a
situation
in
which
one
cannot
avail
oneself
of
this
conjugate
synchronization,
then
it
follows
from
the
distinct,
unrelated
nature
of
the
basepoints
on
either
side
of
the
log-wall
[cf.
the
discussion
of
Remark
3.6.3,
(i)]
that
one
may
only
construct
diagonal
embeddings
of
either
submonoids
of
Galois-
invariants
or
sets
of
Galois-orbits
of
the
various
constant
monoids
[i.e.,
“Ψ
cns
”]
involved.
On
the
other
hand,
such
Galois-invariants
or
Galois-orbits
are
clearly
insufficient
for
conducting
Kummer
theory
[cf.
[IUTchIII],
Remark
1.5.1,
(ii),
for
a
discussion
of
a
related
topic].
Moreover,
the
operation
of
passing
to
sets
of
Galois-orbits
fails
to
be
compatible
with
the
ring
structure
—
e.g.,
the
additive
structure
—
on
[the
formal
union
with
“{0}”
of]
the
various
constant
monoids.
Such
an
incompatibility
is
unacceptable
in
the
context
of
the
theory
of
the
present
series
of
papers
since
it
is
impossible
to
develop
the
theory
of
the
log-wall
[cf.
[AbsTopIII];
[IUTchIII]]
without
applying
the
ring
structure
within
each
Hodge
theater
[cf.
the
discussion
of
Remark
3.6.4,
(i)].
(iii)
As
discussed
at
the
beginning
of
§1,
the
problem
of
giving
an
explicit
description
of
what
one
arithmetic
holomorphic
structure
looks
like
from
the
point
of
view
of
a
distinct
arithmetic
holomorphic
structure
that
is
only
related
to
the
original
arithmetic
holomorphic
structure
via
some
mono-analytic
core
is
one
of
the
central
themes
of
the
theory
of
the
present
series
of
papers.
The
phenomenon
of
conjugate
synchronization
as
discussed
in
(i)
and
(ii)
above,
as
well
as
the
closely
related
phenomenon
of
mono-theta-theoretic
cyclotomic
rigidity
[cf.
the
discussion
of
Remark
3.6.5,
(ii)],
may
be
thought
of
as
particular
instances
of
this
general
theme.
Indeed,
from
the
point
of
view
of
classical
discussions
of
scheme-theoretic
arithmetic
geometry,
the
“natural
isomorphisms”
that
exist
between
various
cyclotomes
that
appear
in
a
discussion
are
typically
taken
for
granted
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
119
—
i.e.,
typically
no
attention
is
given
to
the
issue
of
devising
explicit,
intrinsic
reconstruction
algorithms
for
these
“natural
isomorphisms”
between
cyclotomes.
120
SHINICHI
MOCHIZUKI
Section
4:
Global
Gaussian
Frobenioids
In
the
present
§4,
we
generalize
the
theory
of
Gaussian
monoids,
devel-
oped
in
§3
in
the
case
of
bad
v
∈
V
bad
,
first
to
the
case
of
nonarchimedean
and
archimedean
good
v
∈
V
good
and
then
to
the
global
case.
One
important
feature
of
these
generalizations,
especially
in
the
global
case,
is
the
theme
of
compatibility
with
the
theory
of
ΘNF-
(respectively,
Θ
±ell
-)
Hodge
theaters
—
and,
in
particu-
±
lar,
the
F
l
-
(respectively,
F
l
-)
symmetries
of
such
Hodge
theaters
—
developed
in
[IUTchI],
§4,
§5
(respectively,
[IUTchI],
§6).
In
the
following
discussion,
we
assume
that
we
have
been
given
initial
Θ-
data
as
in
[IUTchI],
Definition
3.1.
We
begin
our
discussion
by
considering
good
nonarchimedean
v
∈
V
good
V
non
.
Proposition
4.1.
(Group-theoretic
Gaussian
Monoids
at
Good
Nonar-
chimedean
Primes)
Let
v
∈
V
good
V
non
.
In
the
notation
of
[IUTchI],
Definition
3.1,
(e),
(f
),
write
def
Π
v
=
Π
−
X
→
v
⊆
Π
±
v
=
Π
X
v
def
⊆
def
Π
cor
=
Π
C
v
v
∼
[cf.
Definition
2.3,
(i),
in
the
case
of
v
∈
V
bad
]
—
so
Π
±
v
/Π
v
=
Z/lZ
[cf.
the
±
∼
±
discussion
preceding
[IUTchI],
Definition
1.1],
Π
cor
v
/Π
v
=
F
l
;
Π
v
G
v
(Π
v
),
±
Π
±
v
G
v
(Π
v
),
cor
Π
cor
v
G
v
(Π
v
)
∼
∼
for
the
quotients
—
which
admit
natural
isomorphisms
G
v
(Π
v
)
→
G
v
(Π
±
v
)
→
cor
∼
±
for
G
v
(Π
v
)
→
G
v
—
determined
by
the
natural
surjections
to
G
v
;
Δ
v
,
Δ
v
,
Δ
cor
v
±
cor
the
respective
kernels
of
these
quotients.
Also,
we
recall
that
Π
v
,
Π
v
,
G
v
(Π
v
),
cor
G
v
(Π
±
v
),
and
G
v
(Π
v
)
may
be
reconstructed
algorithmically
[cf.
[IUTchI],
Corollary
1.2,
and
its
proof;
[AbsAnab],
Lemma
1.3.8]
from
the
topological
group
Π
v
.
(i)
(Constant
Monoids)
The
functorial
group-theoretic
algorithm
of
[Ab-
sTopIII],
Corollary
1.10,
(b)
[cf.
also
the
discussion
of
Remark
1.11.5,
(i),
in
the
case
of
v
∈
V
bad
;
the
discussion
of
“M
v
(−)”
in
[IUTchI],
Definition
5.2,
(v)]
yields
a
functorial
group-theoretic
algorithm
in
the
topological
group
G
v
for
constructing
the
ind-topological
submonoid
[which
is
naturally
isomorphic
to
O
F
]
v
1
Ψ
cns
(G
v
)
⊆
lim
−→
H
(J,
μ
Z
(G
v
))
J
—
where
J
ranges
over
the
open
subgroups
of
G
v
;
μ
Z
(G
v
)
is
as
in
[AbsTopIII],
Corollary
1.10,
(b)
—
equipped
with
its
natural
G
v
-action.
In
particular,
we
obtain
a
functorial
group-theoretic
algorithm
in
the
topological
group
Π
v
for
constructing
the
ind-topological
submonoid
def
1
Ψ
cns
(Π
v
)
=
Ψ
cns
(G
v
(Π
v
))
⊆
lim
−→
H
(G
v
(Π
v
)|
J
,
μ
Z
(G
v
(Π
v
)))
⊆
lim
−→
H
J
1
J
±
(Π
v
|
J
,
μ
Z
(G
v
(Π
v
)))
1
⊆
lim
−→
H
(Π
v
|
J
,
μ
Z
(G
v
(Π
v
)))
J
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
121
—
where
J
ranges
over
the
open
subgroups
of
G
v
(Π
v
)
—
equipped
with
its
natural
G
v
(Π
v
)-action
[cf.
Proposition
3.1,
(ii),
in
the
case
of
v
∈
V
bad
].
(ii)
(Mono-analytic
Semi-simplifications)
The
functorial
algorithm
dis-
cussed
in
[IUTchI],
Example
3.5,
(iii),
for
constructing
“(R
≥0
)
v
”
[cf.
also
[Ab-
sTopIII],
Proposition
5.8,
(iii)]
yields
a
functorial
group-theoretic
algorithm
in
the
topological
group
G
v
for
constructing
a
topological
monoid
R
≥0
(G
v
)
equipped
with
a
natural
isomorphism
∼
×
rlf
Ψ
R
cns
(G
v
)
=
(Ψ
cns
(G
v
)/Ψ
cns
(G
v
)
)
def
→
R
≥0
(G
v
)
—
where
the
superscript
“×”
denotes
the
submonoid
of
units;
the
superscript
“rlf”
denotes
the
realification
[which
is
isomorphic
to
R
≥0
]
of
the
monoid
in
parentheses
[which
is
isomorphic
to
Q
≥0
]
—
and
a
distinguished
element
log
G
v
(p
v
)
∈
R
≥0
(G
v
)
—
i.e.,
the
element
“log
D
Φ
(p
v
)”
of
[IUTchI],
Example
3.5,
(iii).
Write
×
Ψ
ss
cns
(G
v
)
=
Ψ
cns
(G
v
)
×
R
≥0
(G
v
)
def
—
which
we
shall
think
of
as
a
sort
of
“semi-simplified
version”
of
Ψ
cns
(G
v
).
Also,
just
as
in
(i),
we
shall
abbreviate
notation
that
denotes
a
dependence
on
“G
v
(Π
v
)”
[e.g.,
a
“G
v
(Π
v
)”
in
parentheses]
by
means
of
notation
that
denotes
a
dependence
on
“Π
v
”.
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
Let
t
∈
LabCusp
±
(Π
v
)
=
LabCusp
±
(B(Π
v
)
0
)
[cf.
[IUTchI],
Definition
6.1,
(iii)].
In
the
following,
we
shall
use
analogous
conventions
to
the
conventions
introduced
in
Corollary
3.5
concerning
subscripted
labels.
Then
if
we
think
of
the
cuspidal
inertia
groups
⊆
Π
v
corresponding
to
t
as
subgroups
of
cuspidal
inertia
groups
bad
],
then
the
Δ
±
of
Π
±
v
[cf.
Remark
2.3.1,
in
the
case
of
v
∈
V
v
-outer
action
of
±
∼
cor
±
±
F
l
=
Δ
v
/Δ
v
on
Π
v
[cf.
Corollary
2.4,
(iii),
in
the
case
of
v
∈
V
bad
]
induces
isomorphisms
between
the
pairs
def
G
v
(Π
v
)
t
Ψ
cns
(Π
v
)
t
—
consisting
of
a
labeled
ind-topological
monoid
equipped
with
the
action
of
a
labeled
topological
group
—
for
distinct
t
∈
LabCusp
±
(Π
v
).
We
shall
refer
to
these
isomorphisms
as
[F
±
l
-]symmetrizing
isomorphisms
[cf.
Remark
3.5.2,
bad
in
the
case
of
v
∈
V
].
These
symmetrizing
isomorphisms
determine
diagonal
submonoids
Ψ
cns
(Π
v
)
|t|
;
Ψ
cns
(Π
v
)
F
⊆
Ψ
cns
(Π
v
)
|t|
Ψ
cns
(Π
v
)
|F
l
|
⊆
l
|t|∈|F
l
|
|t|∈F
l
of
the
respective
product
monoids
compatible
with
the
respective
actions
by
sub-
scripted
versions
of
G
v
(Π
v
)
[cf.
the
discussion
of
Corollary
3.5,
(i),
in
the
case
of
v
∈
V
bad
],
as
well
as
an
isomorphism
of
ind-topological
monoids
Ψ
cns
(Π
v
)
0
∼
→
Ψ
cns
(Π
v
)
F
l
122
SHINICHI
MOCHIZUKI
compatible
with
the
respective
actions
by
subscripted
versions
of
G
v
(Π
v
)
[cf.
Corol-
lary
3.5,
(iii),
in
the
case
of
v
∈
V
bad
].
(iv)
(Theta
and
Gaussian
Monoids)
Relative
to
the
notational
conventions
discussed
at
the
end
of
(ii),
let
us
write
Ψ
env
(Π
v
)
def
=
Ψ
cns
(Π
v
)
×
×
R
≥0
·
log
Π
v
(p
v
)
·
log
Π
v
(Θ)
—
where
the
notation
“log
Π
v
(p
v
)
·
log
Π
v
(Θ)”
is
to
be
understood
as
a
formal
sym-
bol
[cf.
the
discussion
of
[IUTchI],
Example
3.3,
(ii)]
—
and
Ψ
gau
(Π
v
)
def
=
⊆
Π
v
2
Ψ
cns
(Π
v
)
×
×
R
·
.
.
.
,
j
·
log
(p
),
.
.
.
≥0
v
F
l
Ψ
ss
=
Ψ
cns
(Π
v
)
×
cns
(Π
v
)
j
j
×
R
≥0
(Π
v
)
j
j∈F
l
j∈F
l
—
where,
by
abuse
of
notation,
we
also
write
“j”
for
the
natural
number
∈
{1,
.
.
.
,
l
}
determined
by
an
element
j
∈
F
l
.
In
particular,
[cf.
(i),
(ii),
(iii)]
we
obtain
a
functorial
group-theoretic
algorithm
in
the
topological
group
Π
v
for
construct-
ing
the
theta
monoid
Ψ
env
(Π
v
)
and
the
Gaussian
monoid
Ψ
gau
(Π
v
),
equipped
with
their
[evident]
natural
G
v
(Π
v
)-actions
and
splittings,
as
well
as
the
formal
evaluation
isomorphism
[cf.
Corollary
3.5,
(ii),
in
the
case
of
v
∈
V
bad
]
∼
→
Ψ
gau
(Π
v
)
Π
v
Π
v
Π
v
2
log
(p
v
)
·
log
(Θ)
→
.
.
.
,
j
·
log
(p
v
),
.
.
.
Ψ
env
(Π
v
)
—
which
restricts
to
the
identity
on
the
respective
copies
of
“Ψ
cns
(Π
v
)
×
”
and
is
compatible
with
the
respective
natural
actions
of
G
v
(Π
v
)
as
well
as
with
the
nat-
ural
splittings
on
the
domain
and
codomain.
Proof.
The
various
assertions
of
Proposition
4.1
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.1.1.
(i)
Proposition
4.1
may
be
thought
of
as
a
sort
of
“easy”
formal
general-
ization
of
much
of
the
theory
of
§2,
§3
—
more
precisely,
the
portion
constituted
by
Proposition
3.1
and
Corollaries
2.4,
3.5
—
to
the
case
of
v
∈
V
good
V
non
.
By
comparison
to
the
corresponding
portion
of
the
theory
of
§2,
§3,
this
generalization
is
somewhat
tautological
and,
for
the
most
part,
“vacuous”.
As
we
shall
see
later,
the
reason
for
considering
this
formal
generalization
to
v
∈
V
good
V
non
is
that
it
allows
one
to
“globalize”
the
theory
of
§2,
§3,
i.e.,
by
gluing
together
the
theories
at
v
∈
V
bad
and
v
∈
V
good
.
(ii)
The
symmetrizing
isomorphisms
of
Proposition
4.1,
(iii),
constitute
the
analogue
at
v
∈
V
good
V
non
of
the
conjugate
synchronization
at
v
∈
V
bad
discussed
in
Corollary
3.5,
(i);
Remark
3.5.2.
In
this
context,
it
is
perhaps
most
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
123
natural
to
think
of
the
“copies
of
G
v
(Π
v
)
labeled
by
t
∈
LabCusp
±
(Π
v
)”
as
the
quotients
D
t
/I
t
—
where
I
t
is
a
cuspidal
inertia
group
⊆
Π
v
corresponding
to
t;
D
t
is
the
corresponding
decomposition
group
⊆
Π
v
[i.e.,
the
normalizer,
or,
equivalently,
the
commensurator,
of
I
t
in
Π
v
—
cf.,
e.g.,
[AbsSect],
Theorem
1.3,
(ii)];
we
think
∼
of
D
t
/I
t
as
being
equipped
with
the
isomorphism
D
t
/I
t
→
G
v
(Π
v
)
induced
by
the
natural
surjection
Π
v
G
v
(Π
v
).
(iii)
One
may
also
formulate
an
easy
tautological
formal
analogue
at
v
∈
V
good
V
non
of
the
multiradiality
and
uniradiality
assertions
of
Proposition
3.4,
Corollary
3.7
at
v
∈
V.
For
instance,
(a)
the
construction
of
the
monoids
Ψ
cns
(Π
v
)
[cf.
Proposition
4.1,
(i)]
is
uniradial
[cf.
Proposition
3.4,
(ii)],
while
(b)
the
construction
of
the
monoids
Ψ
ss
cns
(Π
v
),
Ψ
env
(Π
v
),
and
Ψ
gau
(Π
v
)
[cf.
∼
Proposition
4.1,
(ii),
(iv)],
as
well
as
of
the
isomorphism
Ψ
env
(Π
v
)
→
Ψ
gau
(Π
v
)
[cf.
Proposition
4.1,
(iv)],
is
multiradial.
We
leave
the
routine
details
to
the
reader.
Ultimately,
in
the
present
series
of
papers
[cf.,
especially,
the
theory
of
[IUTchIII]],
we
shall
be
interested
in
a
global
analogue
of
the
theory
of
multiradiality
and
uniradiality
developed
in
§1,
§3
at
v
∈
V
bad
.
This
global
analogue
will
“specialize”
to
the
theory
of
§1,
§3
at
v
∈
V
bad
and
to
the
formal
analogue
just
discussed
[i.e.,
(a),
(b)]
at
v
∈
V
good
V
non
.
Proposition
4.2.
(Frobenioid-theoretic
Gaussian
Monoids
at
Good
Nonarchimedean
Primes)
We
continue
to
use
the
notation
of
Proposition
4.1.
Let
†
F
v
be
a
p
v
-adic
Frobenioid
that
appears
in
a
Θ-Hodge
theater
†
HT
Θ
=
({
†
F
w
}
w∈V
,
†
F
mod
)
[cf.
[IUTchI],
Definition
3.6]
—
cf.,
for
instance,
the
Frobe-
nioid
“F
v
=
C
v
”
of
[IUTchI],
Example
3.3,
(i);
here,
we
assume
[for
simplicity]
that
the
base
category
of
†
F
v
is
equal
to
B
temp
(
†
Π
v
)
0
,
and
we
denote
by
means
of
a
“†”
the
various
topological
groups
associated
to
†
Π
v
that
correspond
to
the
topological
groups
associated
to
Π
v
in
Proposition
4.1.
Write
G
v
(
†
Π
v
)
for
the
ind-topological
monoid
Ψ
†
F
v
Ψ
†
F
v
equipped
with
a
continuous
G
v
(
†
Π
v
)-action
determined,
up
to
inner
automorphism
[i.e.,
up
to
an
automorphism
arising
from
an
element
of
†
Π
v
],
by
†
F
v
[cf.
the
construction
of
“Ψ
C
v
”
in
Example
3.2,
(ii),
in
the
case
of
v
∈
V
bad
;
the
discussion
of
“
‡
M
v
”
in
[IUTchI],
Definition
5.2,
(vi);
the
discussion
of
[AbsTopIII],
Remark
3.1.1]
and
†
G
v
Ψ
†
F
v
for
the
ind-topological
monoid
Ψ
†
F
v
equipped
with
a
continuous
†
G
v
-action
deter-
mined,
up
to
inner
automorphism
[i.e.,
up
to
an
automorphism
arising
from
an
124
SHINICHI
MOCHIZUKI
element
of
†
G
v
],
by
the
portion
indexed
by
v
of
the
F
-prime-strip
{
†
F
w
}
w∈V
determined
by
the
Θ-Hodge
theater
†
HT
Θ
[cf.
[IUTchI],
Definition
3.6;
[IUTchI],
Definition
5.2,
(ii)].
(i)
(Constant
Monoids)
There
exists
a
unique
G
v
(
†
Π
v
)-equivariant
iso-
morphism
of
monoids
[cf.
Proposition
3.3,
(ii),
in
the
case
of
v
∈
V
bad
;
the
discussion
of
“
‡
M
v
”
in
[IUTchI],
Definition
5.2,
(vi)]
Ψ
†
F
∼
Ψ
cns
(
†
Π
v
)
→
v
—
cf.
Remark
1.11.1,
(i),
(a);
[AbsTopIII],
Proposition
3.2,
(iv).
(ii)
(Mono-analytic
Semi-simplifications)
There
exists
a
unique
†
G
v
-
equivariant
Z
×
-orbit
of
isomorphisms
of
topological
groups
∼
Ψ
×
†
F
Ψ
cns
(
†
G
v
)
×
→
v
—
cf.
Remark
1.11.1,
(i),
(b);
[AbsTopIII],
Proposition
3.3,
(ii)
—
as
well
as
a
unique
isomorphism
of
monoids
∼
rlf
=
(Ψ
†
F
v
/Ψ
×
Ψ
R
†
F
†
F
)
v
def
†
Ψ
R
cns
(
G
v
)
→
v
that
maps
the
distinguished
element
of
Ψ
R
†
F
determined
by
the
unique
gen-
v
R
†
erator
of
Ψ
†
F
v
/Ψ
×
†
F
to
the
distinguished
element
of
Ψ
cns
(
G
v
)
determined
by
log
†
v
G
v
†
(p
v
)
∈
R
≥0
(
G
v
)
[cf.
Proposition
4.1,
(ii)].
In
particular,
one
may
define
×
R
a
“semi-simplified
version”
Ψ
ss
†
F
=
Ψ
†
F
×
Ψ
†
F
of
Ψ
†
F
;
the
isomorphisms
v
def
v
v
v
discussed
above
determine
a
natural
poly-isomorphism
of
ind-topological
monoids
∼
→
Ψ
ss
†
F
v
†
Ψ
ss
cns
(
G
v
)
[cf.
Proposition
4.1,
(ii)]
that
is
compatible
with
the
natural
splittings
on
the
domain
and
codomain.
Write
Ψ
ss
†
F
def
v
=
Ψ
ss
†
F
;
thus,
it
follows
from
the
definitions
[cf.
also
v
the
unique
isomorphism
of
(i)]
that
we
have
a
natural
isomorphism
[i.e.,
as
opposed
∼
to
a
poly-isomorphism!]
Ψ
ss
→
Ψ
ss
†
F
†
F
.
v
v
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
The
isomorphism
of
(i)
determines,
for
each
t
∈
LabCusp
±
(
†
Π
v
),
a
collection
of
com-
patible
isomorphisms
(Ψ
†
F
)
t
v
∼
→
Ψ
cns
(
†
Π
v
)
t
—
which
are
well-defined
up
to
composition
with
an
inner
automorphism
of
†
Π
v
which
is
independent
of
t
∈
LabCusp
±
(
†
Π
v
)
[cf.
Corollary
3.6,
(i),
in
the
case
of
v
∈
V
bad
]
—
as
well
as
[F
±
l
-]symmetrizing
isomorphisms,
induced
by
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
125
±
∼
†
cor
†
±
the
†
Δ
±
=
Δ
v
/
Δ
v
on
†
Π
±
v
-outer
action
of
F
l
v
[cf.
Corollary
2.4,
(iii),
in
bad
the
case
of
v
∈
V
],
between
the
data
indexed
by
distinct
t
∈
LabCusp
±
(
†
Π
v
).
Moreover,
these
symmetrizing
isomorphisms
determine
[various
diagonal
sub-
monoids,
as
well
as]
an
isomorphism
of
ind-topological
monoids
∼
→
(Ψ
†
F
)
0
v
(Ψ
†
F
)
F
v
l
compatible
with
the
respective
actions
by
subscripted
versions
of
G
v
(
†
Π
v
)
[cf.
Corol-
lary
3.6,
(iii),
in
the
case
of
v
∈
V
bad
].
(iv)
(Theta
and
Gaussian
Monoids)
Write
Ψ
F
gau
(
†
F
v
)
Ψ
†
F
v
Θ
,
for
the
monoids
equipped
with
G
v
(
†
Π
v
)-actions
and
natural
splittings
deter-
mined,
respectively
—
via
the
isomorphisms
of
(i),
(ii),
and
(iii)
—
by
the
monoids
Ψ
env
(
†
Π
v
),
Ψ
gau
(
†
Π
v
),
Galois
actions,
and
splittings
of
Proposition
4.1,
(iv).
Then
the
definition
of
the
various
monoids
involved,
together
with
the
formal
evaluation
isomorphism
of
Proposition
4.1,
(iv),
gives
rise
to
a
collection
of
natural
isomor-
phisms
[cf.
Corollary
3.6,
(ii),
in
the
case
of
v
∈
V
bad
]
Ψ
†
F
v
Θ
∼
→
Ψ
env
(
†
Π
v
)
∼
→
Ψ
gau
(
†
Π
v
)
∼
→
Ψ
F
gau
(
†
F
v
)
—
which
restrict
to
the
identity
or
to
the
[restriction
to
“(−)
×
”
of
the]
isomor-
†
×
phism
of
(i)
[or
its
inverse]
on
the
various
copies
of
Ψ
×
†
F
,
“Ψ
cns
(
Π
v
)
”
and
are
v
compatible
with
the
various
natural
actions
of
G
v
(
†
Π
v
)
and
natural
splittings.
Proof.
The
various
assertions
of
Proposition
4.2
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.2.1.
(i)
In
the
case
of
v
∈
V
bad
treated
in
§3,
we
did
not
discuss
an
analogue
of
the
†
“mono-analytic
semi-simplification”
Ψ
ss
cns
(
G
v
)
of
Proposition
4.1,
(ii).
On
the
other
hand,
one
verifies
immediately
that
one
may
define,
in
the
case
of
v
∈
V
bad
—
via
the
same
group-theoretic
algorithms
as
those
applied
in
Proposition
4.1,
(i),
†
†
†
(ii)
—
ind-topological
monoids
Ψ
ss
cns
(
G
v
),
R
≥0
(
G
v
)
equipped
with
natural
G
v
-
actions,
a
natural
isomorphism
[i.e.,
as
in
the
first
display
of
Proposition
4.1,
(ii)],
†
a
distinguished
element
log
G
v
(p
v
)
∈
R
≥0
(
†
G
v
),
and
a
tautological
splitting
†
Ψ
ss
cns
(
G
v
)
=
†
×
Ψ
ss
×
R
≥0
(
†
G
v
)
cns
(
G
v
)
[cf.
Proposition
4.1,
(ii)].
Moreover,
if
we
write
Ψ
cns
(Π
v
)
def
=
Ψ
cns
(M
Θ
∗
(Π
v
))
126
SHINICHI
MOCHIZUKI
—
where
the
latter
“Ψ
cns
(−)”
is
as
in
Proposition
3.1,
(ii)
—
then,
by
applying
the
cyclotomic
rigidity
isomorphisms
of
Definition
1.1,
(ii),
and
the
discussion
at
the
beginning
of
Corollary
2.9,
one
obtains
a
functorial
group-theoretic
[i.e.,
in
the
topological
group
Π
v
]
Π
v
-equivariant
isomorphism
Ψ
cns
(Π
v
)
×
∼
→
×
Ψ
ss
cns
(G
v
(Π
v
))
good
V
non
in
Proposition
—
cf.
the
discussion
of
“Ψ
ss
cns
(−)”
in
the
case
of
v
∈
V
4.1,
(ii).
Finally,
we
observe
that,
relative
to
the
above
notation,
one
has
analogues
bad
.
We
leave
the
of
“Ψ
ss
†
F
”
and
of
Proposition
4.2,
(i),
(ii),
in
the
case
of
v
∈
V
v
routine
details
to
the
reader.
(ii)
Note
that
in
the
case
of
v
∈
V
good
V
non
,
the
monoids
Ψ
env
(Π
v
),
Ψ
gau
(Π
v
)
of
Proposition
4.1,
(iv),
are
already
divisible.
Thus,
it
is
natural,
in
the
case
of
v
∈
V
good
V
non
,
to
simply
set
∞
Ψ
env
(Π
v
)
∞
Ψ
†
F
v
Θ
def
=
def
=
Ψ
env
(Π
v
);
Ψ
†
F
v
Θ
;
∞
Ψ
gau
(Π
v
)
†
∞
Ψ
F
gau
(
F
v
)
def
=
Ψ
gau
(Π
v
)
def
Ψ
F
gau
(
†
F
v
)
=
—
cf.
the
various
monoids
“
∞
Ψ(−)”
that
appeared
in
the
discussion
of
§3.
(iii)
In
the
situation
of
(ii),
if
one
regards
the
pairs
G
v
(Π
v
)
Ψ
env
(Π
v
),
G
v
(Π
v
)
Ψ
gau
(Π
v
),
G
v
(Π
v
)
∞
Ψ
env
(Π
v
),
G
v
(Π
v
)
∞
Ψ
gau
(Π
v
)
up
to
an
indeterminacy
with
respect
to
Π
v
-inner
automorphisms,
then
one
obtains
data
which
we
shall
denote
by
means
of
the
notation
Ψ
env
(B
temp
(Π
v
)
0
),
Ψ
gau
(B
temp
(Π
v
)
0
),
∞
Ψ
env
(B
temp
(Π
v
)
0
),
∞
Ψ
gau
(B
temp
(Π
v
)
0
)
—
i.e.,
since
Π
v
may
only
be
reconstructed
from
B
temp
(Π
v
)
0
up
to
an
inner
auto-
morphism
indeterminacy
[cf.
the
discussion
of
[IUTchI],
§0].
(iv)
Suppose
that
v
∈
V
bad
.
Then
the
above
discussion
motivates
the
following
notational
conventions.
First,
let
us
write
def
Ψ
gau
(Π
v
)
=
Ψ
gau
(M
Θ
∗
(Π
v
))
def
Θ
∞
Ψ
env
(Π
v
)
=
∞
Ψ
env
(M
∗
(Π
v
)),
def
Θ
∞
Ψ
gau
(Π
v
)
=
∞
Ψ
gau
(M
∗
(Π
v
))
Ψ
env
(Π
v
)
=
Ψ
env
(M
Θ
∗
(Π
v
)),
def
—
cf.
(ii)
above;
the
notation
of
Corollary
3.5,
(ii).
When
these
monoids
equipped
with
various
topological
group
actions
are
considered
only
up
to
a
Π
v
-inner
au-
tomorphism
indeterminacy,
we
shall
denote
the
resulting
data
by
means
of
the
notation
Ψ
env
(B
temp
(Π
v
)
0
),
Ψ
gau
(B
temp
(Π
v
)
0
),
∞
Ψ
env
(B
temp
(Π
v
)
0
),
—
cf.
(iii)
above.
Next,
we
consider
[good]
archimedean
v
∈
V
arc
(⊆
V
good
).
∞
Ψ
gau
(B
temp
(Π
v
)
0
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
127
Proposition
4.3.
(Aut-holomorphic-space-theoretic
Gaussian
Monoids
at
Archimedean
Primes)
Let
v
∈
V
arc
(⊆
V
good
).
Recall
the
Aut-holomorphic
orbispaces
of
[IUTchI],
Example
3.4,
(i),
def
X
v
U
v
=
−
→
U
±
v
=
X
v
def
→
→
def
U
cor
=
C
v
v
∼
—
so
Gal(U
v
/U
±
v
)
=
Z/lZ
[cf.
the
discussion
preceding
[IUTchI],
Definition
1.1],
cor
∼
±
Gal(U
±
v
/U
v
)
=
F
l
;
we
shall
apply
the
notation
“A
”,
“A
”
of
[IUTchI],
Ex-
ample
3.4,
(i),
to
these
Aut-holomorphic
orbispaces.
Also,
we
shall
write
A
⊆
A
⊆
A
for
the
topological
monoid
of
nonzero
elements
of
absolute
value
≤
1
of
the
complex
archimedean
field
A
[cf.
the
slightly
different
notation
of
[AbsTopIII],
Corollary
4.5,
(ii)].
Finally,
we
recall
the
object
D
v
of
the
category
“TM
”
of
split
topological
monoids
discussed
in
[IUTchI],
Example
3.4,
(ii);
we
shall
write
D
v
(U
v
)
when
we
wish
to
regard
D
v
as
an
object
algorithmically
constructed
from
U
v
.
(i)
(Constant
Monoids)
There
is
a
functorial
algorithm
in
the
Aut-
holomorphic
space
U
v
for
constructing
the
topological
monoid
Ψ
cns
(U
v
)
def
=
A
U
v
—
cf.
[IUTchI],
Example
3.4,
(i);
the
discussion
of
“M
v
(−)”
in
[IUTchI],
Defi-
nition
5.2,
(vii);
[AbsTopIII],
Definition
4.1,
(i);
[AbsTopIII],
Corollary
2.7,
(e).
Moreover,
if
we
write
Ψ
cns
(D
v
)
for
the
underlying
topological
monoid
of
D
v
,
then
we
have
a
tautological
isomorphism
of
topological
monoids
Ψ
cns
(U
v
)
∼
→
Ψ
cns
(D
v
(U
v
))
[cf.
[IUTchI],
Example
3.4,
(ii)]
—
which
we
shall
use
to
identify
these
two
topological
monoids.
(ii)
(Mono-analytic
Semi-simplifications)
The
functorial
algorithm
dis-
cussed
in
[IUTchI],
Example
3.5,
(iii),
for
constructing
“(R
≥0
)
v
”
[cf.
also
[Ab-
sTopIII],
Proposition
5.8,
(vi)]
yields
a
functorial
algorithm
in
the
object
D
v
of
TM
for
constructing
a
topological
monoid
R
≥0
(D
v
)
equipped
with
a
distin-
guished
element
D
log
v
(p
v
)
∈
R
≥0
(D
v
)
—
i.e.,
the
element
“log
D
Φ
(p
v
)”
of
[IUTchI],
Example
3.5,
(iii).
Write
×
Ψ
ss
cns
(D
v
)
=
Ψ
cns
(D
v
)
×
R
≥0
(D
v
)
def
—
where
the
superscript
“×”
denotes
the
submonoid
of
units
—
which
we
shall
think
of
as
a
sort
of
“semi-simplified
version”
of
Ψ
cns
(D
v
).
We
shall
abbreviate
notation
that
denotes
a
dependence
on
“D
v
(U
v
)”
[e.g.,
a
“D
v
(U
v
)”
in
parenthe-
ses]
by
means
of
notation
that
denotes
a
dependence
on
“U
v
”.
Finally,
there
is
a
functorial
algorithm
in
the
Aut-holomorphic
space
U
v
for
constructing
the
natural
isomorphism
[which
arises
immediately
from
the
definitions]
×
Ψ
R
cns
(U
v
)
=
Ψ
cns
(U
v
)/Ψ
cns
(U
v
)
def
∼
→
R
≥0
(U
v
)
128
SHINICHI
MOCHIZUKI
—
cf.
[IUTchI],
Example
3.4,
(i).
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
Let
t
∈
LabCusp
±
(U
v
)
[cf.
[IUTchI],
Definition
6.1,
(iii)].
In
the
following,
we
shall
use
analogous
conventions
to
the
conventions
introduced
in
Corollary
3.5
concern-
cor
∼
ing
subscripted
labels.
Then
the
action
of
F
±
=
Gal(U
±
v
/U
v
)
on
the
var-
l
±
ious
Gal(U
v
/U
±
v
)-orbits
of
cusps
of
U
v
[cf.
the
definition
of
“LabCusp
(−)”
in
[IUTchI],
Definition
6.1,
(iii)]
induces
isomorphisms
between
the
labeled
topo-
logical
monoids
Ψ
cns
(U
v
)
t
for
distinct
t
∈
LabCusp
±
(U
v
).
We
shall
refer
to
these
isomorphisms
as
[F
±
l
-
bad
]symmetrizing
isomorphisms
[cf.
Remark
3.5.2,
in
the
case
of
v
∈
V
].
These
symmetrizing
isomorphisms
determine
diagonal
submonoids
Ψ
cns
(U
v
)
|F
l
|
⊆
Ψ
cns
(U
v
)
|t|
;
Ψ
cns
(U
v
)
F
⊆
l
|t|∈|F
l
|
Ψ
cns
(U
v
)
|t|
|t|∈F
l
of
the
respective
product
monoids
[cf.
the
discussion
of
Corollary
3.5,
(i),
in
the
case
of
v
∈
V
bad
],
as
well
as
an
isomorphism
of
topological
monoids
Ψ
cns
(U
v
)
0
∼
→
Ψ
cns
(U
v
)
F
l
[cf.
Corollary
3.5,
(iii),
in
the
case
of
v
∈
V
bad
].
(iv)
(Theta
and
Gaussian
Monoids)
Relative
to
the
notational
conventions
discussed
in
(ii),
let
us
write
Ψ
env
(U
v
)
Ψ
cns
(U
v
)
×
×
def
=
R
≥0
·
log
U
v
(p
v
)
·
log
U
v
(Θ)
—
where
the
notation
“log
U
v
(p
v
)·log
U
v
(Θ)”
is
to
be
understood
as
a
formal
symbol
[cf.
the
discussion
of
[IUTchI],
Example
3.4,
(iii)]
—
and
Ψ
gau
(U
v
)
def
=
⊆
U
v
2
Ψ
cns
(U
v
)
×
×
R
·
.
.
.
,
j
·
log
(p
),
.
.
.
≥0
v
F
l
Ψ
ss
=
Ψ
cns
(U
v
)
×
cns
(U
v
)
j
j
×
R
≥0
(U
v
)
j
j∈F
l
j∈F
l
—
where,
by
abuse
of
notation,
we
also
write
“j”
for
the
natural
number
∈
{1,
.
.
.
,
l
}
determined
by
an
element
j
∈
F
l
.
In
particular,
[cf.
(i),
(ii),
(iii)]
we
obtain
a
functorial
algorithm
in
the
Aut-holomorphic
space
U
v
for
constructing
the
theta
monoid
Ψ
env
(U
v
)
and
the
Gaussian
monoid
Ψ
gau
(U
v
),
equipped
with
their
[ev-
ident]
natural
splittings,
as
well
as
the
formal
evaluation
isomorphism
[cf.
Corollary
3.5,
(ii),
in
the
case
of
v
∈
V
bad
]
∼
→
Ψ
gau
(U
v
)
log
U
v
(p
v
)
·
log
U
v
(Θ)
→
.
.
.
,
j
2
·
log
U
v
(p
v
),
.
.
.
Ψ
env
(U
v
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
129
—
which
restricts
to
the
identity
on
the
respective
copies
of
“Ψ
cns
(U
v
)
×
”
and
is
compatible
with
the
natural
splittings
on
the
domain
and
codomain.
Proof.
The
various
assertions
of
Proposition
4.3
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.3.1.
Analogous
observations
to
the
observations
made
in
Remark
4.1.1,
(i),
(ii),
(iii),
may
be
made
in
the
present
case
of
v
∈
V
arc
.
We
leave
the
rou-
tine
details
to
the
reader.
In
this
context,
we
note
that
the
cuspidal
decomposition
groups
that
appear
in
the
discussion
of
Remark
4.1.1,
(ii),
may
be
thought
of
as
corresponding
to
the
“A
p
”
that
appear
in
[AbsTopIII],
Corollary
2.7,
(e)
—
i.e.,
in
the
construction
of
A
U
v
—
in
the
case
of
points
p
that
belong
to
“sufficiently
small”
neighborhoods
of
the
cusps
that
correspond
to
an
element
t
∈
LabCusp
±
(U
v
).
Proposition
4.4.
(Frobenioid-theoretic
Gaussian
Monoids
at
Archime-
dean
Primes)
We
continue
to
use
the
notation
of
Proposition
4.3.
Let
†
F
v
=
(
†
C
v
,
†
D
v
,
†
κ
v
)
be
the
collection
of
data
indexed
by
v
∈
V
arc
of
a
Θ-Hodge
theater
†
HT
Θ
=
({
†
F
w
}
w∈V
,
†
F
mod
)
[cf.
[IUTchI],
Definition
3.6;
[IUTchI],
Example
3.4,
(i)].
Write
†
F
v
=
(
†
C
v
,
†
D
v
,
†
τ
v
)
for
the
data
indexed
by
v
[cf.
the
discussion
of
[IUTchI],
Example
3.4,
(ii)]
of
the
F
-prime-strip
determined
by
the
Θ-Hodge
theater
†
HT
Θ
[cf.
[IUTchI],
Definition
3.6;
[IUTchI],
Definition
5.2,
(ii)].
Also,
let
def
†
cor
for
the
Aut-holomorphic
orbispaces
associated
us
write
†
U
v
=
†
D
v
and
†
U
±
v
,
U
v
†
±
cor
to
U
v
that
correspond
to
“U
v
”,
“U
v
”,
respectively
[cf.
the
discussion
of
[IUTchI],
Definition
6.1,
(ii)].
(i)
(Constant
Monoids)
In
the
notation
of
[IUTchI],
Definition
3.6;
[IUTchI],
Example
3.4,
(i)
[cf.
also
the
discussion
of
“
‡
M
v
”
in
[IUTchI],
Definition
5.2,
(viii)],
the
Kummer
structure
†
κ
v
:
Ψ
†
F
=
O
(
†
C
v
)
→
A
†
D
v
def
v
on
the
category
†
C
v
,
together
with
the
tautological
equality
†
D
v
=
†
U
v
of
Aut-
holomorphic
spaces,
determine
a
unique
isomorphism
Ψ
†
F
∼
v
→
Ψ
cns
(
†
U
v
)
of
topological
monoids.
def
(ii)
(Mono-analytic
Semi-simplifications)
Write
Ψ
†
F
v
=
O
(
†
C
v
)
[cf.
[IUTchI],
Example
3.4,
(ii)].
Then
there
exists
a
unique
{±1}-orbit
of
isomor-
phisms
of
topological
groups
∼
Ψ
×
†
F
→
v
Ψ
cns
(
†
D
v
)
×
as
well
as
a
unique
isomorphism
of
monoids
Ψ
R
=
Ψ
†
F
v
/Ψ
×
†
F
†
F
v
def
v
∼
→
†
†
Ψ
R
cns
(
D
v
)
=
R
≥0
(
D
v
)
def
130
SHINICHI
MOCHIZUKI
that
maps
the
distinguished
element
of
Ψ
R
†
F
determined
by
p
v
=
e
=
2.71828
.
.
.
v
[i.e.,
the
element
of
the
complex
archimedean
field
that
gives
rise
to
Ψ
†
F
whose
nat-
v
†
ural
logarithm
is
equal
to
1]
to
the
distinguished
element
of
Ψ
R
cns
(
D
v
)
determined
by
log
†
D
v
(p
v
)
∈
R
≥0
(
†
D
v
)
[cf.
the
first
display
of
Proposition
4.3,
(ii)].
In
particular,
×
R
if
we
write
Ψ
ss
†
F
=
Ψ
†
F
×
Ψ
†
F
for
the
“semi-simplified
version”
of
Ψ
†
F
,
v
def
v
v
v
then
the
former
distinguished
element,
together
with
the
poly-isomorphism
of
the
first
display
of
the
present
(ii),
determine
a
natural
poly-isomorphism
of
topological
monoids
∼
†
→
Ψ
ss
Ψ
ss
†
F
cns
(
D
v
)
v
[cf.
Proposition
4.3,
(ii)]
that
is
compatible
with
the
natural
splittings
on
the
domain
and
codomain.
Write
Ψ
ss
†
F
def
v
=
Ψ
ss
†
F
;
thus,
it
follows
from
the
definitions
that
we
v
have
a
natural
isomorphism
Ψ
ss
†
F
∼
v
→
Ψ
ss
†
F
.
v
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
The
isomorphism
of
(i)
determines,
for
each
t
∈
LabCusp
±
(
†
U
v
),
a
collection
of
com-
patible
isomorphisms
∼
→
(Ψ
†
F
)
t
v
Ψ
cns
(
†
U
v
)
t
[cf.
Corollary
3.6,
(i),
in
the
case
of
v
∈
V
bad
],
as
well
as
[F
±
l
-]symmetrizing
±
∼
†
±
†
cor
isomorphisms,
induced
by
the
action
of
F
l
=
Gal(
U
v
/
U
v
)
on
the
vari-
†
†
±
†
ous
Gal(
U
v
/
U
v
)-orbits
of
cusps
of
U
v
[cf.
the
definition
of
“LabCusp
±
(−)”
in
[IUTchI],
Definition
6.1,
(iii)],
between
the
data
indexed
by
distinct
t
∈
LabCusp
±
(
†
U
v
).
Moreover,
these
symmetrizing
isomorphisms
determine
[various
diagonal
sub-
monoids,
as
well
as]
an
isomorphism
of
topological
monoids
∼
→
(Ψ
†
F
)
0
v
(Ψ
†
F
)
F
v
l
[cf.
Corollary
3.6,
(iii),
in
the
case
of
v
∈
V
bad
].
(iv)
(Theta
and
Gaussian
Monoids)
Write
Ψ
F
gau
(
†
F
v
)
Ψ
†
F
v
Θ
,
for
the
topological
monoids
equipped
with
natural
splittings
determined,
respec-
tively
—
via
the
isomorphisms
of
(i),
(ii),
and
(iii)
—
by
the
monoids
Ψ
env
(
†
U
v
),
Ψ
gau
(
†
U
v
)
and
splittings
of
Proposition
4.3,
(iv).
Then
the
definition
of
the
various
monoids
involved,
together
with
the
formal
evaluation
isomorphism
of
Proposition
4.3,
(iv),
gives
rise
to
a
collection
of
natural
isomorphisms
[cf.
Corollary
3.6,
(ii),
in
the
case
of
v
∈
V
bad
]
Ψ
†
F
v
Θ
∼
→
Ψ
env
(
†
U
v
)
∼
→
Ψ
gau
(
†
U
v
)
∼
→
Ψ
F
gau
(
†
F
v
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
131
—
which
restrict
to
the
identity
or
to
the
[restriction
to
“(−)
×
”
of
the]
isomor-
†
×
phism
of
(i)
[or
its
inverse]
on
the
various
copies
of
Ψ
×
†
F
,
“Ψ
cns
(
U
v
)
”
and
are
v
compatible
with
the
various
natural
splittings.
Proof.
The
various
assertions
of
Proposition
4.4
follow
immediately
from
the
definitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.4.1.
In
the
case
of
v
∈
V
arc
,
one
verifies
immediately
that
one
can
make
a
remark
analogous
to
Remark
4.2.1,
(ii).
Corollary
4.5.
(Group-theoretic
Monoids
Associated
to
Base-Θ
±ell
-
Hodge
Theaters)
Let
†
HT
D-Θ
±ell
†
=
(
D
†
±
φ
Θ
±
←−
†
D
T
†
ell
φ
Θ
±
−→
†
D
±
)
be
a
D-Θ
±ell
-Hodge
theater
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
6.4,
(iii)]
and
‡
D
=
{
‡
D
v
}
v∈V
a
D-prime-strip;
here,
we
assume
[for
simplicity]
that
‡
D
v
=
B
temp
(
‡
Π
v
)
0
for
v
∈
V
non
.
Also,
we
shall
denote
the
D
-prime-strip
associated
to
—
i.e.,
the
mono-
analyticization
of
—
a
D-prime-strip
[cf.
[IUTchI],
Definition
4.1,
(iv)]
by
means
of
a
superscript
“”
and
assume
[for
simplicity]
that
‡
D
v
=
B
temp
(
‡
G
v
)
0
for
v
∈
V
non
.
(i)
(Constant
Monoids)
There
is
a
functorial
algorithm
in
the
D-prime-
strip
‡
D
for
constructing
the
assignment
Ψ
cns
(
‡
D)
given
by
def
V
non
v
→
Ψ
cns
(
‡
D)
v
=
G
v
(
‡
Π
v
)
Ψ
cns
(
‡
Π
v
)
V
arc
v
→
Ψ
cns
(
‡
D)
v
=
Ψ
cns
(
‡
D
v
)
def
—
where
the
data
in
brackets
“{−}”
is
to
be
regarded
as
being
well-defined
only
up
to
a
‡
Π
v
-conjugacy
indeterminacy
—
cf.
Remark
4.2.1,
(i),
and
Propositions
3.1,
(ii);
4.1,
(i);
4.3,
(i).
(ii)
(Mono-analytic
Semi-simplifications)
There
is
a
functorial
algo-
‡
rithm
in
the
D
-prime-strip
‡
D
for
constructing
the
assignment
Ψ
ss
cns
(
D
)
given
by
def
non
ss
‡
‡
ss
‡
v
→
Ψ
cns
(
D
)
v
=
G
v
Ψ
cns
(
G
v
)
V
‡
ss
‡
V
arc
v
→
Ψ
ss
cns
(
D
)
v
=
Ψ
cns
(
D
v
)
def
—
where
the
data
in
brackets
“{−}”
is
to
be
regarded
as
being
well-defined
only
up
to
a
‡
G
v
-conjugacy
indeterminacy;
each
“Ψ
ss
cns
(−)”
is
equipped
with
a
splitting,
i.e.,
a
direct
product
decomposition
‡
ss
‡
×
‡
Ψ
ss
cns
(
D
)
v
=
Ψ
cns
(
D
)
v
×
R
≥0
(
D
)
v
132
SHINICHI
MOCHIZUKI
as
the
product
of
the
submonoid
of
units
and
a
submonoid
with
no
nontrivial
units
[each
of
which
is
equipped
with
the
action
of
a
topological
group
when
v
∈
V
non
];
each
submonoid
R
≥0
(
‡
D
)
v
is
equipped
with
a
distinguished
element
‡
log
D
(p
v
)
∈
R
≥0
(
‡
D
)
v
—
cf.
Remark
4.2.1,
(i);
Propositions
4.1,
(ii),
and
4.3,
(ii).
Here,
if
we
regard
‡
D
as
an
object
functorially
constructed
from
‡
D,
then
there
is
a
functorial
algorithm
in
the
D-prime-strip
‡
D
for
constructing
isomorphisms
[of
ind-topological
abelian
groups,
equipped
with
the
action
of
a
topological
group
when
v
∈
V
non
]
∼
Ψ
cns
(
‡
D)
×
v
→
‡
×
Ψ
ss
cns
(
D
)
v
for
each
v
∈
V
—
cf.
Remark
4.2.1,
(i);
Propositions
4.1,
(i),
(ii),
and
4.3,
(i),
(ii).
Finally,
there
is
a
functorial
algorithm
in
the
D
-prime-strip
‡
D
for
constructing
a
Frobenioid
D
(
‡
D
)
”
of
[IUTchI],
Example
3.5,
(iii)]
isomorphic
to
the
Frobe-
[cf.
the
Frobenioid
“D
mod
nioid
“C
mod
”
of
[IUTchI],
Example
3.5,
(i),
equipped
with
a
bijection
∼
Prime(D
(
‡
D
))
→
V
—
where
we
write
“Prime(−)”
for
the
set
of
primes
associated
to
the
divisor
monoid
of
the
Frobenioid
in
parentheses
[cf.
the
discussion
of
[IUTchI],
Exam-
ple
3.5,
(i)]
—
and,
for
each
v
∈
V,
an
isomorphism
of
topological
monoids
∼
‡
ρ
D
,v
:
Φ
D
(
‡
D
),v
→
R
≥0
(
‡
D
)
v
,
where
we
write
“Φ
D
(
‡
D
),v
”
for
the
submonoid
[isomorphic
to
R
≥0
]
of
the
divisor
monoid
of
D
(
‡
D
)
associated
to
v
[cf.
the
iso-
morphism
“ρ
D
v
”
of
[IUTchI],
Example
3.5,
(iii)].
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
Write
†
∼
ζ
:
LabCusp
±
(
†
D
)
→
T
±
for
the
bijection
†
ζ
±
◦
†
ζ
0
Θ
◦
(
†
ζ
0
Θ
)
−1
arising
from
the
bijections
discussed
in
[IUTchI],
Proposition
6.5,
(i),
(ii),
(iii).
Let
t
∈
LabCusp
±
(
†
D
).
In
the
following,
we
shall
use
analogous
conventions
to
the
conventions
introduced
in
Corollary
3.5
concerning
subscripted
labels.
Then
the
various
local
F
±
l
-actions
discussed
in
Corollary
3.5,
(i),
and
Propositions
4.1,
(iii);
4.3,
(iii),
induce
isomorphisms
between
the
labeled
data
Ψ
cns
(
†
D
)
t
ell
[cf.
(i)]
for
distinct
t
∈
LabCusp
±
(
†
D
).
We
shall
refer
to
these
isomorphisms
as
[F
±
l
-]symmetrizing
isomorphisms.
These
symmetrizing
isomorphisms
are
ell
compatible,
relative
to
†
ζ
,
with
the
F
±
l
-symmetry
of
the
associated
D-Θ
-
bridge
[cf.
[IUTchI],
Proposition
6.8,
(i)]
and
determine
diagonal
submonoids
Ψ
cns
(
†
D
)
|F
l
|
⊆
|t|∈|F
l
|
Ψ
cns
(
†
D
)
|t|
;
Ψ
cns
(
†
D
)
F
⊆
l
|t|∈F
l
Ψ
cns
(
†
D
)
|t|
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
133
—
where
the
“⊆’s”
denote
the
various
local
inclusions
of
diagonal
submonoids
of
Corollary
3.5,
(i),
and
Propositions
4.1,
(iii);
4.3,
(iii)
—
as
well
as
an
isomor-
phism
∼
Ψ
cns
(
†
D
)
0
→
Ψ
cns
(
†
D
)
F
l
constituted
by
the
various
corresponding
local
isomorphisms
of
Corollary
3.5,
(iii),
and
Propositions
4.1,
(iii);
4.3,
(iii).
(iv)
(Local
Theta
and
Gaussian
Monoids)
There
is
a
functorial
algo-
rithm
in
the
D-prime-strip
†
D
for
constructing
assignments
Ψ
env
(
†
D
),
Ψ
gau
(
†
D
),
†
†
∞
Ψ
env
(
D
),
∞
Ψ
gau
(
D
)
def
V
v
→
Ψ
env
(
†
D
)
v
=
Ψ
env
(
†
D
,v
);
def
V
v
→
Ψ
gau
(
†
D
)
v
=
Ψ
gau
(
†
D
,v
)
def
V
v
→
∞
Ψ
env
(
†
D
)
v
=
∞
Ψ
env
(
†
D
,v
)
def
V
v
→
∞
Ψ
gau
(
†
D
)
v
=
∞
Ψ
gau
(
†
D
,v
)
—
where
the
various
local
data
are
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[for
all
v
∈
V],
as
described
in
detail
in
Corollary
3.5,
(ii),
(iii),
and
Propositions
4.1,
(iv);
4.3,
(iv)
[cf.
also
Remarks
4.2.1,
(ii),
(iii),
(iv);
4.4.1]
—
as
well
as
compatible
evaluation
isomorphisms
Ψ
env
(
†
D
)
∼
→
Ψ
gau
(
†
D
);
†
∞
Ψ
env
(
D
)
∼
→
†
∞
Ψ
gau
(
D
)
as
described
in
detail
in
Corollary
3.5,
(ii),
and
Propositions
4.1,
(iv);
4.3,
(iv).
(v)
(Global
Realified
Theta
and
Gaussian
Frobenioids)
There
is
a
func-
torial
algorithm
in
the
D
-prime-strip
†
D
for
constructing
a
Frobenioid
†
D
env
(
D
)
—
namely,
as
a
copy
of
the
Frobenioid
“D
(
†
D
)”
of
(ii)
above,
multiplied
by
a
†
formal
symbol
“log
D
(Θ)”
[cf.
the
constructions
of
Propositions
4.1,
(iv),
and
4.3,
(iv),
as
well
as
of
[IUTchI],
Example
3.5,
(ii)]
—
isomorphic
to
the
Frobenioid
†
”
of
[IUTchI],
Example
3.5,
(i),
equipped
with
a
bijection
Prime(D
env
(
D
))
“C
mod
∼
→
V
[cf.
(ii)
above]
and,
for
each
v
∈
V,
an
isomorphism
of
topological
monoids
∼
→
Ψ
env
(
†
D
)
R
Φ
D
env
(
†
D
),v
v
—
where
we
write
“Φ
D
env
(
†
D
),v
”
for
the
submonoid
[isomorphic
to
R
≥0
]
of
the
†
divisor
monoid
of
D
env
(
D
)
associated
to
v;
we
write
Ψ
env
(
†
D
)
R
v
for
the
data
†
[which,
as
is
easily
verified,
is
completely
determined
by
D
—
cf.
Propositions
4.1,
(ii),
(iv),
and
4.3,
(ii),
(iv),
as
well
as
the
evident
analogues
of
these
results
at
bad
primes,
i.e.,
in
the
spirit
of
Remark
4.2.1,
(i)]
obtained
from
Ψ
env
(
†
D
)
v
[cf.
(iv)
above]
by
replacing
the
ind-topological
monoid
portion
of
Ψ
env
(
†
D
)
v
by
the
realification
of
the
quotient
of
this
ind-topological
monoid
by
its
submonoid
of
units.
There
is
a
functorial
algorithm
in
the
D
-prime-strip
†
D
for
constructing
a
subcategory,
equipped
with
a
Frobenioid
structure,
(
†
D
)
⊆
D
(
†
D
)
j
D
gau
j∈F
l
134
SHINICHI
MOCHIZUKI
—
[cf.
Remark
4.5.2,
(i),
below
concerning
the
subscript
“j’s”]
whose
divisor
and
rational
function
monoids
are
determined
[relative
to
the
divisor
and
rational
func-
tion
monoids
of
each
factor
in
the
product
category
of
the
display]
by
the
“vector
of
ratios”
2
.
.
.
,
j
·,
.
.
.
whose
components
are
indexed
by
j
∈
F
l
[cf.
Remark
4.5.4
below;
the
nota-
tional
conventions
of
Propositions
4.1,
(iv);
4.3,
(iv)]
—
equipped
with
a
bijection
∼
(
†
D
))
→
V
[cf.
(ii)
above]
and,
for
each
v
∈
V,
an
isomorphism
of
Prime(D
gau
topological
monoids
∼
→
Ψ
gau
(
†
D
)
R
Φ
D
gau
(
†
D
),v
v
—
where
we
write
“Φ
D
gau
(
†
D
),v
”
for
the
submonoid
[isomorphic
to
R
≥0
]
of
the
divisor
monoid
of
D
gau
(
†
D
)
associated
to
v;
we
write
Ψ
gau
(
†
D
)
R
v
for
the
data
†
[which,
as
is
easily
verified,
is
completely
determined
by
D
—
cf.
Propositions
4.1,
(ii),
(iv),
and
4.3,
(ii),
(iv),
as
well
as
the
evident
analogues
of
these
results
at
bad
primes,
i.e.,
in
the
spirit
of
Remark
4.2.1,
(i)]
obtained
from
Ψ
gau
(
†
D
)
v
[cf.
(iv)
above]
by
replacing
the
ind-topological
monoid
portion
of
Ψ
gau
(
†
D
)
v
by
the
realification
of
the
quotient
of
this
ind-topological
monoid
by
its
submonoid
of
units.
Finally,
there
is
a
functorial
algorithm
in
the
D
-prime-strip
†
D
for
constructing
a
global
formal
evaluation
isomorphism
of
Frobenioids
†
(
D
)
D
env
∼
→
D
gau
(
†
D
)
which
is
compatible,
relative
to
the
bijections
and
local
isomorphisms
of
topological
monoids
associated
to
these
Frobenioids,
with
the
local
evaluation
isomorphisms
of
(iv)
above.
Proof.
The
various
assertions
of
Corollary
4.5
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.5.1.
(i)
Just
as
was
done
in
Definition
3.8,
one
may
interpret
the
various
collections
of
monoids
constructed
in
Corollary
4.5,
(i),
(iv)
as
collections
of
Frobenioids.
That
is
to
say,
the
collection
of
monoids
discussed
in
Corollary
4.5,
(i),
gives
rise
to
an
F-prime-strip,
hence
also
to
an
associated
F
-prime-strip.
In
a
similar
vein,
the
theta
and
Gaussian
monoids
of
Corollary
4.5,
(iv),
give
rise
to
a
well-defined
F
-
prime-strip
—
up
to
an
indeterminacy,
at
the
v
∈
V
bad
[corresponding
to
the
various
2l-th
roots
of
the
square
of
the
theta
function
and
“value-profiles”],
relative
to
automorphisms
of
the
split
Frobenioid
at
such
v
∈
V
bad
that
induce
the
identity
automorphism
on
the
subcategory
of
isometries
[cf.
[FrdI],
Theorem
5.1,
(iii)]
of
the
underlying
category
of
the
split
Frobenioid
—
cf.
Remark
4.10.1
below.
On
the
other
hand,
as
discussed
in
Remark
3.8.1,
this
Frobenioid-theoretic
formulation
is
—
by
comparison
to
the
original
monoid-theoretic
formulation
—
technically
ill-suited
to
discussions
of
conjugate
synchronization.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
135
(ii)
On
the
other
hand,
such
technical
complications
do
not
occur
if
one
re-
stricts
oneself
to
discussions
of
realifications
—
cf.,
e.g.,
the
objects
“R
≥0
(
‡
D
)
v
”,
“D
(
‡
D
)”
discussed
in
Corollary
4.5,
(ii).
In
general,
Frobenioid-theoretic
formu-
lations
are
typically
technically
easier
to
work
with
than
monoid-theoretic
formula-
tions
when
the
associated
“Picard
groups
P
ic
Φ
(−)”
[cf.
[FrdI],
Theorem
5.1;
[FrdI],
Theorem
6.4,
(i);
[IUTchI],
Remark
3.1.5]
contain
nontorsion
elements
—
i.e.,
at
a
more
intuitive
level,
when
there
is
a
nontrivial
notion
of
the
“degree”
of
a
line
bundle.
Examples
of
such
Frobenioids
include
global
arithmetic
Frobenioids
such
as
the
Frobenioid
“D
(
‡
D
)”
of
Corollary
4.5,
(ii),
as
well
as
the
tempered
Frobenioids
that
appeared
in
Propositions
3.3
and
3.4;
Corollary
3.6.
Remark
4.5.2.
(i)
One
may
also
construct
symmetrizing
isomorphisms
as
in
Corollary
4.5,
(iii),
for
versions
labeled
by
t
∈
LabCusp
±
(
†
D
)
of
the
semi-simplifications
†
Ψ
ss
cns
(
D
),
equipped
with
splittings
and
distinguished
elements,
and
the
global
re-
alified
Frobenioids
D
(
†
D
),
equipped
with
bijections
and
local
isomorphisms
of
topological
monoids,
as
discussed
in
Corollary
4.5,
(iii).
We
leave
the
routine
de-
tails
to
the
reader.
(ii)
Just
as
was
discussed
in
Remark
3.5.3,
one
may
also
consider
“multi-
basepoint”
versions
of
the
symmetrizing
isomorphisms
of
Corollary
4.5,
(iii)
[cf.
also
the
discussion
of
(i)
above]
—
i.e.,
by
passing
to
D-Θ
ell
-bridges
or
[holomorphic
or
mono-analytic]
capsules
or
processions
[cf.
[IUTchI],
Proposition
6.8,
(i),
(ii),
(iii);
[IUTchI],
Proposition
6.9,
(i),
(ii)].
We
leave
the
routine
details
to
the
reader.
Remark
4.5.3.
Before
proceeding,
we
pause
to
review
the
significance
of
the
F
±
l
-symmetry
that
gives
rise
to
the
symmetrizing
isomorphisms
of
Corollary
4.5,
(iii)
[cf.
Remark
3.5.2].
(i)
First,
we
recall
that
the
crucial
conjugate
synchronization
established
in
Corollaries
3.5,
(i);
4.5,
(iii)
[cf.
also
Propositions
4.1,
(iii);
4.3,
(iii)],
is
possible
in
the
case
of
the
F
±
l
-symmetry
—
but
not
in
the
case
of
the
F
l
-symmetry!
—
precisely
because
of
the
connectedness,
at
each
v
∈
V,
of
the
local
components
involved
—
cf.
the
discussion
of
Remarks
2.6.1,
(i);
2.6.2,
(i);
3.5.2,
(ii),
as
well
as
[IUTchI],
Remark
6.12.4,
(i),
(ii).
Here,
we
note
in
passing
that
although
these
remarks
essentially
only
concern
v
∈
V
bad
,
similar
[but,
in
some
sense,
easier!]
remarks
hold
at
v
∈
V
good
.
A
related
property
of
the
F
±
l
-symmetry
—
which,
again,
is
not
satisfied
by
the
F
l
-symmetry!
—
is
the
“geometric”
nature
of
the
automorphisms
that
give
rise
to
this
symmetry
[cf.
Remark
3.5.2,
(iii)].
(ii)
One
way
to
understand
the
significance
of
the
“single
basepoint”
sym-
metrizing
isomorphisms
arising
from
the
F
±
l
-symmetry
is
to
compare
these
sym-
metrizing
isomorphisms
with
the
symmetrizing
isomorphisms
that
arise
from
the
various
“multi-basepoint”
[i.e.,
“multi-connected
component”]
symmetries
discussed
in
Remarks
3.5.3;
4.5.2,
(ii).
That
is
to
say:
(a)
By
comparison
to
the
symmetries
that
arise
from
mono-analytic
cap-
sules/processions:
the
ring
structure
—
i.e.,
“arithmetic
holomorphic
136
SHINICHI
MOCHIZUKI
structure”
—
that
remains
intact
in
the
case
of
the
symmetrizing
isomor-
phisms
of
Corollary
4.5,
(iii),
will
play
an
essential
role
in
the
theory
of
the
log-wall
[cf.
the
discussion
of
Remark
3.6.4,
(i)],
which
we
shall
apply
in
[IUTchIII].
(b)
By
comparison
to
the
symmetries
that
arise
from
holomorphic
cap-
sules/processions:
the
“single
basepoint”
that
remains
intact
in
the
case
of
the
symmetrizing
isomorphisms
of
Corollary
4.5,
(iii),
is
used
not
only
to
establish
conjugate
synchronization,
but
also
to
maintain
a
bijective
link
with
the
set
of
labels
in
“LabCusp
±
(−)”
[cf.
the
discussion
of
Re-
mark
3.5.2].
Both
conjugate
synchronization
and
the
bijective
link
with
the
set
of
labels
play
crucial
roles
in
the
theory
of
Galois-theoretic
theta
evaluation
developed
in
§3
[cf.
the
various
remarks
following
Corollaries
3.5,
3.6;
Remark
3.8.3].
(c)
By
comparison
to
the
symmetries
that
arise
from
the
F
±
l
-symmetries
of
ell
ell
D-Θ
-bridges:
Although
the
structure
of
a
D-Θ
-bridge
allows
one
to
maintain
a
bijective
link
with
the
set
of
labels
in
“LabCusp
±
(−)”
[cf.
the
discussion
of
[IUTchI],
Remark
4.9.2,
(i);
[IUTchI],
Remark
6.12.4,
(i)],
ell
the
multi-basepoint
nature
of
the
F
±
l
-symmetries
of
D-Θ
-bridges
does
not
allow
one
to
establish
conjugate
synchronization
[cf.
(b)].
(iii)
Note
that
in
order
to
glue
together
the
various
local
F
±
l
-symmetries
of
Corollary
3.5,
(i),
and
Propositions
4.1,
(iii);
4.3,
(iii),
so
as
to
obtain
the
global
F
±
l
-symmetry
of
Corollary
4.5,
(iii),
it
is
necessary
to
make
use
of
the
global
portion
“
†
D
±
”
of
the
D-Θ
±ell
-Hodge
theater
under
consideration
—
i.e.,
by
ap-
plying
the
theory
of
[IUTchI],
Proposition
6.5
[cf.
also
[IUTchI],
Remark
6.12.4,
(iii)].
That
is
to
say,
the
global
portion
of
the
D-Θ
±ell
-Hodge
theater
under
con-
sideration
plays,
in
particular,
the
role
of
synchronizing
the
±-indeterminacies
at
each
v
∈
V.
Indeed,
in
some
sense,
this
is
precisely
the
content
of
[IUTchI],
Proposition
6.5.
In
particular,
the
essential
role
played
in
this
context
by
“
†
D
±
”
in
synchronizing,
or
coordinating,
the
various
local
±-indeterminacies
is
one
important
underlying
cause
for
the
profinite
conjugacy
indeterminacies
—
i.e.,
“
Δ”-conjugacy
in-
determinacies
—
that
occur
in
Corollaries
2.4,
2.5
—
cf.
the
discussion
of
Remark
2.5.2.
Thus,
in
summary,
these
local
±-indeterminacies
constitute
one
important
reason
for
the
need
to
apply
the
“complements
on
tempered
coverings”
developed
in
[IUTchI],
§2,
in
the
proof
of
Corollary
2.4
of
the
present
paper.
Remark
4.5.4.
In
the
situation
of
Corollary
4.5,
(v),
we
remark
that
the
Frobe-
(
†
D
)
may
be
thought
of
as
a
sort
of
“weighted
diagonal”,
relative
to
nioid
D
gau
the
weights
determined
by
the
vector
“(.
.
.
,
j
2
·,
.
.
.
)”.
That
is
to
say,
at
a
more
concrete
level,
the
divisor
monoid
(respectively,
rational
function
monoid)
of
this
Frobenioid
consists
of
elements
of
the
form
(1
2
·
φ,
2
2
·
φ,
.
.
.
,
j
2
·
φ,
.
.
.
)
(respectively,
(1
2
·
β,
2
2
·
β,
.
.
.
,
j
2
·
β,
.
.
.
))
—
where
φ
(respectively,
β)
is
an
element
of
the
divisor
monoid
(respectively,
rational
function
monoid)
associated
to
the
Frobenioid
D
(
†
D
).
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
137
Corollary
4.6.
(Frobenioid-theoretic
Monoids
Associated
to
Θ
±ell
-
Hodge
Theaters)
Let
†
HT
Θ
±ell
†
=
(
F
†
±
Θ
ψ
±
←−
†
F
T
†
ell
Θ
ψ
±
−→
†
D
±
)
be
a
Θ
±ell
-Hodge
theater
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
6.11,
(iii)]
and
‡
F
=
{
‡
F
v
}
v∈V
an
F-prime-strip;
here,
we
assume
[for
simplicity]
that
the
D-Θ
±ell
-Hodge
theater
±ell
[cf.
[IUTchI],
Definition
6.11,
(iii)]
is
the
D-Θ
±ell
-Hodge
associated
to
†
HT
Θ
±ell
of
Corollary
4.5,
and
that
the
D-prime-strip
associated
to
‡
F
theater
†
HT
D-Θ
[cf.
[IUTchI],
Remark
5.2.1,
(i)]
is
the
D-prime-strip
‡
D
of
Corollary
4.5.
Also,
we
shall
denote
the
F
-prime-strip
associated
to
—
i.e.,
the
mono-analyticization
of
—
an
F-prime-strip
[cf.
[IUTchI],
Definition
5.2.1,
(ii)]
by
means
of
a
superscript
“”.
(i)
(Constant
Monoids)
There
is
a
functorial
algorithm
in
the
F-prime-
strip
‡
F
for
constructing
the
assignment
Ψ
cns
(
‡
F)
given
by
def
non
‡
‡
V
v
→
Ψ
cns
(
F)
v
=
G
v
(
Π
v
)
Ψ
‡
F
v
V
arc
v
→
Ψ
cns
(
‡
F)
v
=
Ψ
‡
F
v
def
—
where
the
data
in
brackets
“{−}”
is
to
be
regarded
as
being
well-defined
only
up
to
a
‡
Π
v
-conjugacy
indeterminacy
—
cf.
[IUTchI],
Definition
5.2,
(i);
Propo-
sitions
3.3,
(ii)
[i.e.,
where
we
take
“
†
C
v
”
to
be
‡
F
v
];
4.2,
(i);
4.4,
(i).
We
shall
write
∼
Ψ
cns
(
‡
F)
→
Ψ
cns
(
‡
D)
for
the
collection
of
isomorphisms
of
data
indexed
by
v
∈
V
determined
by
the
“Kummer-theoretic”
isomorphisms
of
Propositions
3.3,
(ii)
[i.e.,
where
we
take
“
†
C
v
”
to
be
‡
F
v
and
apply
the
conventions
discussed
in
Remark
4.2.1.,
(i);
cf.
also
Proposition
1.3,
(ii),
(iii)];
4.2,
(i);
4.4,
(i).
(ii)
(Mono-analytic
Semi-simplifications)
There
is
a
functorial
algo-
‡
rithm
in
the
F
-prime-strip
‡
F
for
constructing
the
assignment
Ψ
ss
cns
(
F
)
given
by
def
‡
ss
V
v
→
Ψ
ss
cns
(
F
)
v
=
Ψ
‡
F
v
—
where
we
regard
each
“Ψ
ss
‡
F
”
as
being
equipped
with
its
natural
splitting
and,
v
when
v
∈
V
non
,
its
associated
distinguished
element;
for
v
∈
V
non
,
“Ψ
ss
‡
F
”
is
v
to
be
regarded
as
being
well-defined
only
up
to
a
†
G
v
-conjugacy
indeterminacy
—
cf.
Remark
4.2.1,
(i),
and
Propositions
4.2,
(ii);
4.4,
(ii).
We
shall
write
‡
Ψ
ss
cns
(
F
)
∼
→
‡
Ψ
ss
cns
(
D
)
for
the
collection
of
isomorphisms
of
data
indexed
by
v
∈
V
determined
by
the
“Kummer-theoretic”
isomorphisms
of
Propositions
4.2,
(ii);
4.4,
(ii)
—
cf.
also
Remark
4.2.1,
(i);
Corollary
4.5,
(ii).
Now
recall
the
F
-prime-strip
‡
F
∼
=
(
‡
C
,
Prime(
‡
C
)
→
V,
‡
F
,
{
‡
ρ
v
}
v∈V
)
138
SHINICHI
MOCHIZUKI
associated
to
‡
F
in
[IUTchI],
Remark
5.2.1,
(ii).
Then,
in
the
notation
of
Corollary
4.5,
(ii);
[IUTchI],
Remark
5.2.1,
(ii),
there
is
an
isomorphism
of
Frobenioids
‡
C
∼
D
(
‡
D
)
→
that
is
uniquely
determined
by
the
condition
that
it
be
compatible
with
the
∼
respective
bijections
Prime(−)
→
V
and
local
isomorphisms
of
topologi-
cal
monoids
for
each
v
∈
V,
relative
to
the
above
collection
of
isomorphisms
‡
∼
ss
‡
Ψ
ss
cns
(
F
)
→
Ψ
cns
(
D
).
Finally,
there
is
a
functorial
algorithm
for
construct-
∼
ing
from
the
F
-prime-strip
‡
F
[recalled
above]
the
isomorphism
‡
C
→
D
(
‡
D
)
[of
the
preceding
display]
and
the
[necessarily
compatible]
collection
of
isomorphisms
‡
∼
ss
‡
Ψ
ss
cns
(
F
)
→
Ψ
cns
(
D
)
[cf.
Remark
4.6.1
below].
(iii)
(Labels,
F
±
l
-Symmetries,
and
Conjugate
Synchronization)
In
the
notation
of
Corollary
4.5,
(iii),
the
collection
of
isomorphisms
of
(i)
determines,
for
each
t
∈
LabCusp
±
(
†
D
),
a
collection
of
compatible
isomorphisms
∼
Ψ
cns
(
†
F
)
t
→
Ψ
cns
(
†
D
)
t
—
where
the
†
Π
v
-conjugacy
indeterminacy
at
each
v
∈
V
non
[cf.
(i)]
is
in-
dependent
of
t
∈
LabCusp
±
(
†
D
)
—
as
well
as
[F
±
l
-]symmetrizing
isomor-
±
phisms,
induced
by
the
various
local
F
l
-actions
discussed
in
Corollary
3.6,
(i),
and
Propositions
4.2,
(iii);
4.4,
(iii),
between
the
data
indexed
by
distinct
t
∈
LabCusp
±
(
†
D
).
Moreover,
these
symmetrizing
isomorphisms
are
compat-
ible,
relative
to
†
ζ
[cf.
Corollary
4.5,
(iii)],
with
the
F
±
l
-symmetry
of
the
ell
associated
D-Θ
-bridge
[cf.
[IUTchI],
Proposition
6.8,
(i)]
and
determine
[various
diagonal
submonoids,
as
well
as]
an
isomorphism
∼
Ψ
cns
(
†
F
)
0
→
Ψ
cns
(
†
F
)
F
l
constituted
by
the
various
corresponding
local
isomorphisms
of
Corollary
3.6,
(iii),
and
Propositions
4.2,
(iii);
4.4,
(iii).
(iv)
(Local
Theta
and
Gaussian
Monoids)
Let
(
†
F
J
†
Θ
ψ
−→
†
F
>
†
HT
Θ
)
be
a
Θ-bridge
[relative
to
the
given
initial
Θ-data
—
cf.
[IUTchI],
Definition
5.5,
(ii)]
which
is
glued
to
the
Θ
±
-bridge
associated
to
the
Θ
±ell
-Hodge
theater
±ell
†
HT
Θ
via
the
functorial
algorithm
of
[IUTchI],
Proposition
6.7
[so
J
=
T
]
—
cf.
the
discussion
of
[IUTchI],
Remark
6.12.2,
(i).
Then
there
is
a
functo-
rial
algorithm
in
the
Θ-bridge
of
the
above
display,
equipped
with
its
gluing
to
±ell
the
Θ
±
-bridge
associated
to
†
HT
Θ
,
for
constructing
assignments
Ψ
F
env
(
†
HT
Θ
),
Ψ
F
gau
(
†
HT
Θ
),
∞
Ψ
F
env
(
†
HT
Θ
),
∞
Ψ
F
gau
(
†
HT
Θ
)
[where
we
make
a
slight
abuse
of
the
notation
“
†
HT
Θ
”]
def
V
v
→
Ψ
F
env
(
†
HT
Θ
)
v
=
Ψ
†
F
v
Θ
;
def
V
v
→
Ψ
F
gau
(
†
HT
Θ
)
v
=
Ψ
F
gau
(
†
F
v
)
def
V
v
→
∞
Ψ
F
env
(
†
HT
Θ
)
v
=
∞
Ψ
†
F
v
Θ
def
V
v
→
∞
Ψ
F
gau
(
†
HT
Θ
)
v
=
∞
Ψ
F
gau
(
†
F
v
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
139
—
where
the
various
local
data
are
equipped
with
actions
by
topological
groups
when
v
∈
V
non
and
splittings
[for
all
v
∈
V],
as
described
in
detail
in
Corollary
3.6,
(ii),
(iii),
and
Propositions
4.2,
(iv);
4.4,
(iv)
[cf.
also
Remarks
4.2.1,
(ii);
4.4.1]
—
as
well
as
compatible
evaluation
isomorphisms
Ψ
F
env
(
†
HT
Θ
)
∼
→
∼
Θ
†
∞
Ψ
F
env
(
HT
)
→
Ψ
env
(
†
D
>
)
→
∼
Ψ
gau
(
†
D
>
)
∼
†
∞
Ψ
env
(
D
>
)
→
∼
†
∞
Ψ
gau
(
D
>
)
Ψ
F
gau
(
†
HT
Θ
);
→
∼
→
Θ
†
∞
Ψ
F
gau
(
HT
)
as
described
in
detail
in
Corollary
3.6,
(ii)
[cf.
also
Remark
4.2.1,
(iv);
the
left-hand
portion
of
the
first
display
of
Proposition
3.4,
(i);
the
first
display
of
Proposition
3.7,
(i)],
and
Propositions
4.2,
(iv);
4.4,
(iv)
[cf.
also
Corollary
4.5,
(iv)].
(v)
(Global
Realified
Theta
and
Gaussian
Frobenioids)
By
applying
—
i.e.,
in
the
fashion
of
the
constructions
of
Propositions
4.2,
(iv);
4.4,
(iv)
—
both
labeled
[as
in
(iii)
—
cf.
Remark
4.6.2,
(ii),
below]
and
non-labeled
versions
of
the
∼
†
(
D
)”,
isomorphism
“
‡
C
→
D
(
‡
D
)”
of
(ii)
to
the
global
Frobenioids
“D
env
†
“D
gau
(
D
)”
constructed
in
Corollary
4.5,
(v),
one
obtains
a
functorial
algo-
rithm
in
the
Θ-bridge
of
the
first
display
of
(iv),
equipped
with
its
gluing
to
the
±ell
Θ
±
-bridge
associated
to
†
HT
Θ
,
for
constructing
Frobenioids
†
(
HT
Θ
),
C
env
†
C
gau
(
HT
Θ
)
—
where
again
we
make
a
slight
abuse
of
the
notation
“
†
HT
Θ
”;
we
note
in
passing
†
that
the
construction
of
“C
env
(
HT
Θ
)”
is
essentially
similar
to
the
construction
of
†
“C
tht
”
in
[IUTchI],
Example
3.5,
(ii)
—
together
with
bijections
Prime(C
env
(
HT
Θ
))
∼
∼
†
→
V,
Prime(C
gau
(
HT
Θ
))
→
V
and
isomorphisms
of
topological
monoids
∼
∼
Φ
C
env
→
Ψ
F
env
(
†
HT
Θ
)
R
(
†
HT
Θ
),v
v
;
Φ
C
gau
→
Ψ
F
gau
(
†
HT
Θ
)
R
(
†
HT
Θ
),v
v
[cf.
the
notational
conventions
of
Corollary
4.5,
(v)]
for
each
v
∈
V,
as
well
as
evaluation
isomorphisms
∼
†
(
HT
Θ
)
C
env
→
†
D
env
(
D
>
)
∼
→
D
gau
(
†
D
>
)
∼
→
†
C
gau
(
HT
Θ
)
—
i.e.,
in
the
fashion
of
the
constructions
of
Propositions
4.2,
(iv);
4.4,
(iv),
by
“conjugating”
the
evaluation
isomorphism
of
Corollary
4.5,
(v),
by
the
isomorphism
∼
“
‡
C
→
D
(
‡
D
)”
of
(ii)
—
which
are
compatible,
relative
to
the
local
iso-
morphisms
of
topological
monoids
for
each
v
∈
V
discussed
above,
with
the
local
evaluation
isomorphisms
of
(iv).
Proof.
The
various
assertions
of
Corollary
4.6
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.6.1.
One
verifies
easily
that,
in
the
case
of
v
∈
V
non
,
the
poly-
∼
ss
†
isomorphism
Ψ
ss
also
Remark
4.2.1,
†
F
→
Ψ
cns
(
G
v
)
of
Proposition
4.2,
(ii)
[cf.
v
(i)],
may
be
reconstructed
algorithmically
from
†
F
v
.
By
contrast,
in
the
case
of
v
∈
V
arc
,
it
is
not
possible
to
reconstruct
algorithmically
[the
non-unit
portion
of]
140
SHINICHI
MOCHIZUKI
∼
ss
the
corresponding
poly-isomorphism
Ψ
ss
†
F
→
Ψ
cns
(D
v
)
of
Proposition
4.4,
(ii),
from
†
v
F
v
.
That
is
to
say,
in
the
case
of
v
∈
V
arc
,
the
distinguished
element
of
Ψ
ss
†
F
[i.e.,
v
†
of
Ψ
R
F
v
.
On
the
other
hand,
†
F
]
is
not
preserved
by
arbitrary
automorphisms
of
v
∼
‡
ss
‡
in
the
context
of
Corollary
4.6,
(ii),
if
one
reconstructs
both
Ψ
ss
cns
(
F
)
→
Ψ
cns
(
D
)
‡
∼
‡
and
C
→
D
(
D
)
in
a
compatible
fashion,
then
the
distinguished
elements
at
v
∈
V
arc
may
be
computed
[in
the
evident
fashion]
from
the
distinguished
elements
at
v
∈
V
non
,
together
with
the
structure
of
the
global
Frobenioids
‡
C
,
D
(
‡
D
),
i.e.,
by
thinking
of
these
global
Frobenioids
as
“devices
for
currency
exchange”
between
the
various
“local
currencies”
constituted
by
the
divisor
monoids
at
the
various
v
∈
V
[cf.
[IUTchI],
Remark
3.5.1,
(ii)].
Remark
4.6.2.
(i)
Similar
observations
to
the
observations
made
in
Remark
4.5.1,
(i),
con-
cerning
the
content
of
Corollary
4.5,
(i),
(iv),
may
be
made
in
the
case
of
Corollary
4.6,
(i),
(iv).
(ii)
Similar
observations
to
the
observations
made
in
Remark
4.5.2,
(i),
(ii),
concerning
the
content
of
Corollary
4.5,
(iii),
may
be
made
in
the
case
of
Corollary
4.6,
(iii).
Corollary
4.7.
(Group-theoretic
Monoids
Associated
to
Base-ΘNF-
Hodge
Theaters)
Let
†
HT
D-ΘNF
=
(
†
D
†
φ
NF
←−
†
D
J
†
φ
Θ
−→
†
D
>
)
be
a
D-ΘNF-Hodge
theater
[cf.
[IUTchI],
Definition
4.6,
(iii)]
which
is
glued
±ell
to
the
D-Θ
±ell
-Hodge
theater
†
HT
D-Θ
of
Corollary
4.5
via
the
functorial
al-
gorithm
of
[IUTchI],
Proposition
6.7
[so
J
=
T
]
—
cf.
the
discussion
of
[IUTchI],
Remark
6.12.2,
(i),
(ii).
(i)
(Non-realified
Global
Structures)
There
is
a
functorial
algorithm
in
the
category
†
D
for
constructing
the
morphism
†
D
→
†
D
[i.e.,
a
“category-theoretic
version”
of
the
natural
morphism
of
hyperbolic
orbicurves
C
K
→
C
F
mod
]
of
[IUTchI],
Example
5.1,
(i),
the
monoid/field/pseudo-monoid
equipped
with
natural
π
1
(
†
D
)-/(π
1
rat
(
†
D
)
)π
1
κ-sol
(
†
D
)-actions
π
1
(
†
D
)
M
(
†
D
),
π
1
(
†
D
)
M
(
†
D
),
π
1
κ-sol
(
†
D
)
M
(
†
D
)
∞
κ
—
which
are
well-defined
up
to
π
1
(
†
D
)-/π
1
κ-sol
(
†
D
)-conjugacy
indetermina-
cies
—
of
[IUTchI],
Example
5.1,
(i),
the
submonoids/subfield/subset
of
π
1
(
†
D
)-
rat/κ-sol
†
/π
1
(
D
)-/π
1
κ-sol
(
†
D
)-invariants
†
M
mod
(
D
)
⊆
(π
1
κ-sol
(
†
D
)
)
†
M
sol
(
D
)
⊆
M
(
†
D
),
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
M
mod
(
†
D
)
⊆
M
(
†
D
),
†
M
κ
(
D
)
⊆
141
M
(
†
D
)
∞
κ
[cf.
[IUTchI],
Example
5.1,
(i)],
the
[“corresponding”]
Frobenioids
(
†
D
)
F
mod
F
(
†
D
)
⊆
F
(
†
D
)
←
(
†
D
),
F
(
†
D
)
for
the
categories
“
†
F
mod
”,
“
†
F
”
ob-
—
where
we
write
F
mod
tained
in
[IUTchI],
Example
5.1,
(iii),
by
taking
the
“
†
F
”
of
loc.
cit.
to
be
F
(
†
D
),
and,
by
abuse
of
notation,
we
regard
the
Frobenioid
F
mod
(
†
D
)
as
being
equipped
with
a
natural
bijection
Prime(F
mod
(
†
D
))
∼
→
V
[cf.
the
final
portion
of
[IUTchI],
Example
5.1,
(v)]
—
of
[IUTchI],
Example
5.1,
(ii),
(iii),
and
the
natural
realification
functor
(
†
D
)
F
mod
→
R
†
F
mod
(
D
)
[cf.
[IUTchI],
Example
5.1,
(vii);
[FrdI],
Proposition
5.3].
(ii)
(Labels
and
F
l
-Symmetry)
Recall
the
bijection
†
∼
ζ
:
LabCusp(
†
D
)
→
J
∼
(
→
F
l
)
of
[IUTchI],
Proposition
4.7,
(iii).
In
the
following,
we
shall
use
analogous
conven-
tions
to
the
conventions
applied
in
Corollary
4.5
concerning
subscripted
labels.
Let
j
∈
LabCusp(
†
D
).
Then
there
is
a
functorial
algorithm
in
the
category
†
D
for
constructing
an
F-prime-strip
F
(
†
D
)|
j
—
which
is
well-defined
up
to
isomorphism
—
from
F
(
†
D
)
[cf.
[IUTchI],
Example
5.4,
(iv),
where
we
take
the
“δ”
of
loc.
cit.
to
be
j].
Moreover,
the
natural
†
poly-action
of
F
l
on
D
[cf.
[IUTchI],
Example
4.3,
(iv)]
induces
isomorphisms
between
the
labeled
data
F
(
†
D
)|
j
,
†
M
mod
(
D
)
j
,
†
{π
1
κ-sol
(
†
D
)
M
sol
(
D
)}
j
,
M
mod
(
†
D
)
j
,
{π
1
κ-sol
(
†
D
)
M
(
†
D
)}
j
,
∞
κ
R
†
(
†
D
)
j
→
F
mod
(
D
)
j
F
mod
[cf.
(i)]
for
distinct
j
∈
LabCusp(
†
D
)
[cf.
Remark
4.7.2
below].
We
shall
refer
to
these
isomorphisms
as
[F
l
-]symmetrizing
isomorphisms.
Here,
the
objects
rat
†
equipped
with
π
1
(
D
)(
π
1
κ-sol
(
†
D
))-actions
are
to
be
regarded
as
being
subject
rat/κ-sol
†
to
independent
π
1
(
D
)-conjugacy
indeterminacies
for
distinct
j,
to-
rat
†
gether
with
a
single
(π
1
(
D
)
)π
1
κ-sol
(
†
D
)-conjugacy
indeterminacy
that
is
independent
of
j
[cf.
the
discussion
of
the
final
portion
of
[IUTchI],
Exam-
ple
5.1,
(i)].
These
symmetrizing
isomorphisms
are
compatible,
relative
to
†
ζ
,
with
the
F
l
-symmetry
of
the
associated
D-NF-bridge
[cf.
[IUTchI],
Proposition
142
SHINICHI
MOCHIZUKI
4.9,
(i)]
and
determine
diagonal
F-prime-strips/submonoids/subrings/sub-
pseudo-monoids
[equipped
with
a
group
action
subject
to
conjugacy
indetermina-
cies
as
described
above]/subcategories
[cf.
Remark
4.7.2
below]
(−)
F
l
⊆
(−)
j
j∈F
l
—
where
“(−)
...
”
may
be
taken
to
be
F
(
†
D
)|
...
[cf.
the
discussion
of
[IUTchI],
†
†
κ-sol
†
†
Example
5.4,
(i)],
M
(
D
)
M
mod
(
D
)
...
,
M
mod
(
D
)
...
,
{π
1
sol
(
D
)}
...
,
R
†
{π
1
κ-sol
(
†
D
)
M
(
†
D
)}
...
,
F
mod
(
†
D
)
...
,
or
F
mod
(
D
)
...
[cf.
the
discus-
∞
κ
sion
of
[IUTchI],
Example
5.1,
(vii)].
[Here,
the
notion
of
a
“diagonal
F-prime-
strip”,
of
a
“diagonal
sub-pseudo-monoid
equipped
with
a
group
action
subject
to
conjugacy
indeterminacies
as
described
above”,
or
of
a
“diagonal
subcategory”
is
to
be
understood
in
a
purely
formal
sense,
i.e.,
as
a
purely
formal
notational
shorthand
for
the
F
l
-symmetrizing
isomorphisms
discussed
above.]
(iii)
(Localization
Functors
and
Realified
Global
Structures)
Let
j
∈
LabCusp(
†
D
).
For
simplicity,
write
†
D
j
=
{
†
D
v
j
}
v∈V
,
†
D
j
=
{
†
D
v
j
}
v∈V
for
the
D-,
D
-prime-strips
associated
[cf.
[IUTchI],
Definition
4.1,
(iv);
[IUTchI],
Remark
5.2.1,
(i)]
to
the
F-prime-strip
F
(
†
D
)|
j
.
Then
there
is
a
functorial
algorithm
in
the
category
†
D
for
constructing
[1-]compatible
collections
of
“lo-
calization”
functors/poly-morphisms
[up
to
isomorphism]
(
†
D
)
j
F
mod
→
R
†
F
mod
(
D
)
j
F
(
†
D
)|
j
,
{π
1
κ-sol
(
†
D
)
M
(
†
D
)}
j
∞
κ
→
→
(F
(
†
D
)|
j
)
R
M
∞
κv
(
†
D
v
j
)
⊆
M
∞
κ×v
(
†
D
v
j
)
v∈V
—
where
the
superscript
“R”
denotes
the
realification
—
as
in
the
discussion
of
[IUTchI],
Example
5.4,
(iv),
(vi)
[cf.
also
[IUTchI],
Definition
5.2,
(v),
(vii)],
together
with
a
natural
isomorphism
of
Frobenioids
D
(
†
D
j
)
∼
→
R
†
F
mod
(
D
)
j
[cf.
the
notation
of
Corollary
4.5,
(ii)]
and,
for
each
v
∈
V,
a
natural
isomorphism
of
topological
monoids
R
≥0
(
†
D
j
)
v
∼
→
Ψ
(F
(
†
D
)|
j
)
R
,v
—
where
“Ψ
(F
(
†
D
)|
j
)
R
,v
”
denotes
the
divisor
monoid
associated
to
the
Frobe-
nioid
that
constitutes
(F
(
†
D
)|
j
)
R
at
v
—
which
are
compatible
[cf.
Remark
4.7.1
below]
with
the
respective
bijections
involving
“Prime(−)”
and
the
respective
R
†
local
isomorphisms
of
topological
monoids
[cf.
the
arrow
F
mod
(
D
)
j
→
†
R
(F
(
D
)|
j
)
discussed
above;
Corollary
4.5,
(ii)].
Finally,
all
of
these
structures
are
compatible
with
the
respective
F
l
-symmetrizing
isomorphisms
[cf.
(ii)].
Proof.
The
various
assertions
of
Corollary
4.7
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
143
Remark
4.7.1.
Similar
observations
to
the
observations
made
in
Remark
4.5.2,
(i),
(ii),
concerning
the
F
±
l
-symmetrizing
isomorphisms
of
Corollary
4.5,
(iii),
may
be
made
in
the
case
of
the
F
l
-symmetrizing
isomorphisms
of
Corollary
4.7,
(ii).
Remark
4.7.2.
In
the
context
of
Corollary
4.7,
(ii),
we
recall
from
Remarks
3.5.2,
(iii);
4.5.3,
(i),
that
unlike
the
case
with
F
±
l
-symmetry,
in
the
case
of
F
l
-
symmetry,
it
is
not
possible
to
establish
the
sort
of
conjugate
synchronization
given
in
Corollary
4.5,
(iii),
since
the
F
l
-symmetry
involves
—
i.e.,
more
precisely,
arises
from
conjugation
by
elements
with
nontrivial
image
in
—
the
arithmetic
portion
[i.e.,
the
absolute
Galois
group
of
the
base
field]
of
the
global
arithmetic
fundamental
groups
involved
[cf.
the
discussion
of
how
“G
K
-conjugacy
indeter-
minacies
give
rise
to
G
v
-conjugacy
indeterminacies”
in
Remark
2.5.2,
(iii)].
It
is
precisely
this
state
of
affairs
that
obliges
us,
in
Corollary
4.7,
(ii),
to
work
with
(a)
F-prime-strips,
as
opposed
to
the
corresponding
ind-topological
monoids
with
Galois
actions
as
in
Corollary
4.5,
(iii),
and
with
(b)
the
various
objects
introduced
in
Corollary
4.7,
(i),
that
are
equipped
with
sub-/super-scripts
“mod”,
“sol”,
“κ-sol”,
or
“
∞
κ”
—
corresponding
to
“F
mod
”,
“F
sol
”,
“π
1
κ-sol
(−)”,
or
“
∞
κ-coric
rational
rat/κ-sol
functions”
—
or
[as
in
the
case
of
“π
1
(−)”]
are
only
defined
up
to
certain
conjugacy
indeterminacies,
as
opposed
to
the
objects
not
equipped
with
such
subscripts
or
not
subject
to
such
conjugacy
indeter-
minacies.
That
is
to
say,
both
(a)
and
(b)
allow
one
to
ignore
the
various
independent
—
i.e.,
non-synchronizable
—
conjugacy
indeterminacies
that
occur
at
the
various
distinct
labels
as
a
consequence
of
the
single
basepoint
with
respect
to
which
one
consid-
ers
both
the
labels
and
the
labeled
objects
[cf.
the
discussion
of
Remark
3.5.2,
(ii)].
Here,
it
is
also
useful
to
observe
that
by
working
with
the
various
objects
introduced
in
Corollary
4.7,
(i),
that
are
equipped
with
a
sub-/super-script
“mod”,
“sol”,
or
“κ-sol”
—
i.e.,
on
which
the
various
conjugacy
indeterminacies
involved
act
in
a
synchronized
fashion
—
one
may
construct
the
various
diagonal
subcategories
as-
sociated
to
the
corresponding
Frobenioids
in
a
fashion
in
which
one
is
not
obliged
to
contend
with
the
technical
subtleties
that
arise
from
independent
conjugacy
indeterminacies
at
distinct
labels
[cf.
the
discussion
of
“Galois-invariants/Galois-
orbits”
in
Remark
3.8.3,
(ii)].
In
[IUTchIII],
the
ring
structure
on
these
objects
equipped
with
a
subscript
“mod”
will
be
applied
as
a
sort
of
translation
appara-
tus
between
“
-line
bundles”
[i.e.,
arithmetic
line
bundles
thought
of
as
additive
modules
with
additional
structure]
and
“
-line
bundles”
[i.e.,
arithmetic
line
bun-
dles
thought
of
“multiplicatively”
or
“idèlically”,
as
in
the
theory
of
Frobenioids]
—
cf.
[AbsTopIII],
Definition
5.3,
(i),
(ii).
Remark
4.7.3.
At
this
point,
it
is
of
interest
to
review
the
significance
of
the
F
±
l
-
and
F
l
-symmetries
in
the
context
of
the
theory
of
the
present
§4.
(i)
First,
we
recall
that,
in
the
context
of
the
present
series
of
papers,
the
“F
l
”
that
appears
in
the
notation
“F
±
l
”
and
“F
l
”
is
to
be
thought
of
—
since
l
is
144
SHINICHI
MOCHIZUKI
“large”
—
as
a
sort
of
finite
approximation
of
the
ring
of
rational
integers
Z
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.3,
(i)].
That
is
to
say,
the
F
±
l
-symmetry
corresponds
to
the
additive
structure
of
Z,
while
the
F
l
-symmetry
corresponds
to
the
multiplicative
structure
of
Z.
Since
the
“F
l
”
under
consideration
arises
from
the
torsion
points
of
an
elliptic
curve,
it
is
natural
—
especially
in
light
of
the
central
role
played
in
the
present
series
of
papers
by
v
∈
V
bad
—
to
think
of
the
“Z”
under
consideration
as
the
Galois
group
“Z”
of
the
universal
combinatorial
covering
of
the
Tate
curves
that
appear
at
v
∈
V
bad
[cf.
the
discussion
at
the
beginning
of
[EtTh],
§1].
In
particular,
in
light
of
the
theory
of
Tate
curves,
it
is
natural
to
think
of
this
“Z”
as
representing
a
sort
of
universal
version
of
the
value
group
associated
to
a
local
field
that
occurs
at
a
v
∈
V
bad
,
and
to
think
of
the
element
0
∈
Z
—
hence,
the
label
0
∈
|F
l
|
—
as
representing
the
units.
(ii)
Perhaps
the
most
fundamental
difference
between
the
F
±
l
-
and
F
l
-sym-
metries
lies
in
the
fact
that
the
F
±
l
-symmetry
involves
the
zero
label
0
∈
|F
l
|
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.5].
In
particular,
the
F
±
l
-symmetry
is
suited
to
application
to
the
“units”
—
i.e.,
to
the
various
local
“O
×
”
and
“O
×μ
”
that
appear
in
the
theory.
At
a
more
technical
level,
this
relationship
between
the
×
F
±
l
-symmetry
and
“O
”
may
be
seen
in
the
theory
of
§3
[cf.
also
Corollaries
4.5,
(iii);
4.6,
(iii)].
That
is
to
say,
in
§3
[cf.
the
discussion
of
Remark
3.8.3],
the
F
±
l
-
symmetry
is
applied
precisely
to
establish
conjugate
synchronization,
which,
in
turn,
will
be
applied
eventually
to
establish
the
crucial
coricity
of
“O
×μ
”
in
the
context
of
the
Θ
×μ
gau
-link
[cf.
Corollary
4.10,
(iv),
below].
Here,
let
us
observe
that
the
conjugate
synchronization,
established
by
means
of
the
F
±
l
-symmetry,
of
copies
of
the
absolute
Galois
group
of
the
local
base
field
at
various
v
∈
V
non
is
a
very
delicate
property
that
depends
quite
essentially
on
the
“arithmetic
holomorphic
structure”
of
the
Hodge
theaters
under
consideration.
That
is
to
say,
from
the
point
of
view
of
the
theory
of
§1,
conjugate
synchronization
in
one
Hodge
theater
fails
to
be
compatible
with
conjugate
synchronization
in
another
Hodge
theater
with
a
distinct
arithmetic
holomorphic
structure.
Put
another
way,
from
the
point
of
view
of
the
theory
of
§1,
conjugate
synchronization
can
only
be
naturally
formulated
in
a
uniradial
fashion.
This
uniradiality
may
also
be
seen
at
a
purely
combinatorial
level,
as
we
shall
discuss
in
Remark
4.7.4
below.
On
the
other
hand,
if
one
passes
to
mono-analyticizations
—
e.g.,
to
mono-analytic
processions
—
then
the
mono-
analytic
“O
×μ
”
that
appears
in
the
Θ
×μ
gau
-link
[cf.
Corollary
4.10,
(iv),
below]
is,
by
contrast,
coric.
That
is
to
say,
by
relating
the
zero
label,
which
is
common
to
distinct
arithmetic
holomorphic
structures,
to
the
various
nonzero
labels,
which
belong
to
a
single
fixed
arithmetic
holomorphic
structure,
the
condition
of
invariance
with
respect
to
the
F
±
l
-symmetry
may
—
e.g.,
in
the
case
of
the
mono-analytic
“O
×μ
”
—
amount
to
a
condition
of
coricity.
In
particular,
in
the
case
of
the
mono-analytic
“O
×μ
”,
the
F
±
l
-symmetry
plays
the
role
of
establishing
the
coric
pieces
—
i.e.,
components
which
are
“uniform”
with
respect
to
all
of
the
distinct
arith-
metic
holomorphic
structures
involved
—
of
the
apparatus
to
be
estab-
lished
in
the
present
series
of
papers.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
145
This
dual
role
—
i.e.,
consisting
of
both
uniradial
and
coric
aspects
—
played
by
the
F
±
l
-symmetry
is
to
be
considered
in
contrast
to
the
strictly
multiradial
role
[cf.
(iii)
below]
played
by
the
F
l
-symmetry.
Also,
in
this
context,
we
observe
that
the
symmetrization,
effected
by
the
F
±
l
-symmetry,
between
zero
and
nonzero
labels
may
be
thought
of,
from
the
point
of
view
of
(i),
as
a
symmetrization
between
[local]
units
and
value
groups
and,
hence,
in
particular,
is
reminiscent
of
the
intertwining
of
units
and
value
groups
effected
by
the
log-link
[cf.
[IUTchIII],
Remark
3.12.2,
(i),
(ii)],
as
well
as
of
the
crucial
compatibility
between
the
F
±
l
-
symmetrizing
isomorphisms
[i.e.,
that
give
rise
to
the
conjugate
synchronization]
and
the
log-link
[cf.
[IUTchIII],
Remark
1.3.2].
(iii)
The
significance
of
the
F
l
-symmetry
lies,
in
a
word,
in
the
fact
that
it
allows
one
to
separate
the
zero
label
from
the
nonzero
labels.
From
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
this
property
makes
the
F
l
-
symmetry
well-suited
for
the
construction/description
of
the
internal
structure
of
the
Gaussian
monoids,
which
are,
in
effect,
“distributions”
or
“functions”
of
a
parameter
j
∈
F
l
[cf.
Corollaries
4.5,
(iv),
(v);
4.6,
(iv),
(v)].
Here,
we
note
that
this
separation
of
the
zero
label
—
which
parametrizes
coric
data
that
is
common
to
distinct
arithmetic
holomorphic
structures
—
from
the
nonzero
labels
—
which
parametrize
the
components
of
the
Gaussian
monoid
associated
to
a
particular
arithmetic
holomorphic
structure
—
is
crucial
from
the
point
of
view
of
describing
the
Gaussian
monoid
associated
to
a
particular
arithmetic
holomorphic
structure
in
terms
that
may
be
understood
from
the
point
of
view
of
some
“alien”
arithmetic
holomorphic
structure.
Put
another
way,
from
the
point
of
view
of
the
theory
of
§1,
the
F
l
-symmetry
admits
a
natural
multiradial
formulation.
This
multiradiality
may
also
be
seen
at
a
purely
combinatorial
level,
as
we
shall
discuss
in
Remark
4.7.4
below.
In
this
context,
it
is
important
to
note
that
if
one
thinks
of
the
coric
constant
distribution,
labeled
by
zero,
as
embedded
via
the
diagonal
embedding
into
the
various
products
parametrized
by
j
∈
F
l
that
appear
in
the
construction
of
the
Gaussian
monoids
[cf.
the
isomorphisms
that
appear
in
the
final
displays
of
Corollaries
4.5,
(iii);
4.6,
(iii)],
then
it
is
natural
to
think
of
the
volumes
computed
at
each
j
∈
F
l
as
being
assigned
a
weight
1/l
—
i.e.,
so
that
the
diagonal
embedding
of
the
constant
distribution
is
compatible
with
taking
the
constant
distribution
to
be
of
weight
1
[cf.
the
discussion
of
[IUTchI],
Remark
5.4.2].
Put
another
way,
from
the
point
of
view
of
“computation
of
weighted
volumes”,
the
various
nonzero
j
∈
F
l
are
“subordinate”
to
0
∈
|F
l
|
—
i.e.,
F
l
j
≪
0.
In
particular,
to
symmetrize,
in
the
context
of
the
internal
structure
of
the
Gaussian
monoids,
the
zero
and
nonzero
labels
[i.e.,
as
in
the
case
of
the
F
±
l
-symmetry!]
amounts
to
allowing
a
relation
“0
≪
0”
—
which
is
absurd
[i.e.,
in
the
sense
that
it
fails
to
be
compatible
with
weighted
volume
computations]!
Remark
4.7.4.
(i)
One
way
to
understand
the
underlying
combinatorial
structure
of
the
uniradiality
of
the
F
±
l
-symmetry
and
the
multiradiality
of
the
F
l
-symmetry
[cf.
the
discussion
of
Remark
4.7.3,
(ii),
(iii)]
is
to
consider
these
symmetries
—
which
146
SHINICHI
MOCHIZUKI
are
defined
relative
to
some
given
arithmetic
holomorphic
structure
[or,
at
a
more
technical
level,
some
given
Θ
±ell
NF-Hodge
theater
—
cf.
[IUTchI],
Definition
6.13,
(i)]
—
in
the
context
of
the
étale-pictures
that
arise
from
each
of
these
symmetries
[cf.
[IUTchI],
Corollaries
4.12,
6.10].
In
the
case
of
the
F
±
l
-
(respectively,
F
l
-)
symmetry,
this
étale-picture
consists
of
a
collection
of
copies
of
F
l
(respectively,
|F
l
|
=
F
{0}),
each
copy
corresponding
to
a
single
arithmetic
holomorphic
l
structure,
which
are
glued
together
at
the
coric
label
0
∈
F
l
(respectively,
0
∈
|F
l
|).
In
Fig.
4.1
(respectively,
4.2)
below,
an
illustration
is
given
of
such
an
étale-picture,
in
which
the
notation
“±”
(respectively,
“”)
is
used
to
denote
the
various
elements
of
F
l
\
{0}
(respectively,
F
l
)
in
each
copy
of
F
l
(respectively,
|F
l
|).
Moreover,
on
each
copy
of
F
l
(respectively,
|F
l
|)
—
labeled,
say,
by
some
spoke
α
[corresponding
to
a
single
arithmetic
holomorphic
structure]
—
one
has
a
natural
action
of
a
“corresponding
copy”
of
F
±
(respectively,
F
l
l
).
(ii)
The
fundamental
difference
between
the
simple
combinatorial
models
of
the
étale-pictures
considered
in
(i)
lies
in
the
fact
that
whereas
±
(a)
in
the
case
of
the
F
±
l
-symmetry,
the
F
l
-actions
on
distinct
spokes
fail
to
commute
with
one
another,
(b)
in
the
case
of
the
F
l
-symmetry,
the
F
l
-actions
on
distinct
spokes
com-
mute
with
one
another
and,
moreover,
are
compatible
with
the
permu-
tations
of
spokes
discussed
in
[IUTchI],
Corollary
4.12,
(iii).
Indeed,
the
noncommutativity,
or
“incompatibility
with
simultaneous
execution
at
distinct
spokes”
[cf.
Remark
1.9.1],
of
(a)
is
a
direct
consequence
of
the
inclusion
of
the
zero
label
in
the
F
±
l
-symmetry
and
may
be
thought
of
as
a
sort
of
pro-
totypical
combinatorial
representation
of
the
phenomenon
of
uniradiality.
By
contrast,
the
commutativity,
or
“compatibility
with
simultaneous
execution
at
distinct
spokes”
[cf.
Remark
1.9.1],
of
(b)
is
a
direct
consequence
of
the
exclusion
of
the
zero
label
from
the
F
l
-symmetry
and
may
be
thought
of
as
a
sort
of
prototypi-
cal
combinatorial
representation
of
the
phenomenon
of
multiradiality.
Note
that
in
the
case
of
the
F
±
l
-symmetry,
it
is
also
a
direct
consequence
of
the
inclusion
of
the
zero
label
that
the
condition
of
invariance
with
respect
to
the
F
±
l
-actions
on
all
of
the
spokes
may
be
thought
of
as
a
condition
of
“uniformity”
among
the
elements
of
the
copies
of
F
l
at
the
various
spokes,
hence
as
a
sort
of
coricity
[cf.
the
discussion
of
Remark
4.7.3,
(ii)].
±
±
±
±
...
...
↓↑
±
±
±
±
→
←
0
←
→
±
±
±
±
Fig.
4.1:
Étale-picture
of
F
±
l
-symmetries
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
...
147
...
|
—
0
—
Fig.
4.2:
Étale-picture
of
F
l
-symmetries
(iii)
Although
the
combinatorial
versions
of
uniradiality
and
multiradiality
dis-
cussed
in
(ii)
above
are
not
formulated
in
terms
of
the
formalism
of
uniradial
and
multiradial
environments
developed
in
§1
[cf.
Example
1.7,
(ii)],
it
is
not
difficult
to
produce
such
a
formulation.
For
instance,
one
may
take
the
coric
data
to
consist
of
objects
of
the
form
“0
α
”
—
i.e.,
the
zero
label,
subscripted
by
the
label
α
associated
to
some
spoke.
For
any
two
spokes
α,
β,
we
define
the
set
of
arrows
0
α
→
0
β
to
consist
of
precisely
one
element
(α,
β).
We
then
take,
in
the
case
of
the
F
±
l
-
-)
symmetry,
the
radial
data
to
consist
of
a
copy
(F
)
(respectively,
(respectively,
F
l
α
l
|F
l
|
α
)
of
F
l
(respectively,
|F
l
|)
subscripted
by
the
label
α
associated
to
some
spoke.
For
any
two
spokes
α,
β,
we
define
the
set
of
arrows
(F
l
)
α
→
(F
l
)
β
(respectively,
|F
l
|
α
→
|F
l
|
β
)
to
consist
of
precisely
one
element
if
the
actions
(F
±
l
)
γ
(F
l
)
γ
(respectively,
(F
)
(|F
|)
),
for
γ
=
α,
β,
determine
an
action
of
l
γ
l
γ
±
(respectively,
(F
(F
±
l
)
α
×
(F
l
)
β
l
)
α
×
(F
l
)
β
)
on
the
co-product
(F
l
)
α
0
(F
l
)
β
(respectively,
(|F
l
|)
α
0
(|F
l
|)
β
)
obtained
by
identifying
the
respective
zero
labels
0
α
,
0
β
,
and
to
equal
the
empty
set
if
such
an
action
does
not
exist.
Then
one
has
a
natural
radial
functor
(F
l
)
α
→
0
α
(respectively,
|F
l
|
α
→
0
α
)
that
associates
coric
data
to
radial
data.
Moreover,
the
resulting
radial
environment
is
easily
seen
to
be
uniradial
(respectively,
multiradial).
We
leave
the
routine
details
to
the
reader.
Finally,
we
note
in
passing
that
the
formulation
involving
products
given
above
is
reminiscent
both
of
the
discussion
of
the
switching
functor
in
Example
1.7,
(iii),
and
of
the
discussion
of
parallel
transport
via
connections
in
Remark
1.7.1.
Remark
4.7.5.
In
the
context
of
the
discussion
of
the
combinatorial
models
±
of
the
F
l
-
and
F
l
-symmetries
in
Remark
4.7.4,
it
is
useful
to
recall
that
the
F
±
l
-
and
F
l
-symmetries
correspond,
respectively,
to
the
additive
and
multiplicative
structures
of
the
field
F
l
—
which
[cf.
Remark
4.7.3,
(i)]
we
wish
to
think
of
as
a
sort
of
finite
approximation
of
the
ring
Z.
That
is
to
say,
from
the
point
of
view
of
the
theory
of
the
present
series
of
papers,
(a)
the
F
±
l
-
and
F
l
-symmetries
correspond,
respectively,
to
the
two
com-
binatorial
dimensions
—
i.e.,
addition
and
multiplication
—
of
a
ring
[cf.
the
discussion
of
[AbsTopIII],
§I3].
148
SHINICHI
MOCHIZUKI
Moreover,
in
the
context
of
the
discussion
of
Remark
4.7.3,
(i),
concerning
units
and
value
groups,
it
is
useful
to
recall
that
these
two
combinatorial
dimensions
may
be
thought
of
as
corresponding
to
(b)
the
units
and
value
group
of
a
mixed-characteristic
nonarchimedean
or
complex
archimedean
local
field
[cf.
the
discussion
of
[AbsTopIII],
§I3]
or,
alternatively,
to
(c)
the
two
cohomological
dimensions
of
the
absolute
Galois
group
of
a
mixed-characteristic
nonarchimedean
local
field
or
the
two
underlying
real
dimensions
of
a
complex
archimedean
local
field
[cf.
the
discussion
of
[AbsTopIII],
§I3].
Finally,
the
hierarchical
structure
of
these
two
dimensions
—
i.e.,
the
way
in
which
“one
dimension
[i.e.,
multiplication]
is
piled
on
top
of
the
other
[i.e.,
addition]”
—
is
reflected
in
the
(d)
subordination
structure
“≪”,
relative
to
the
computation
of
weighted
volumes,
of
nonzero
labels
with
respect
to
the
zero
label
[cf.
the
discussion
of
Remark
4.7.3,
(iii)].
as
well
as
in
the
fact
that
(e)
the
F
±
l
-symmetry
arises
from
the
conjugation
action
of
the
geometric
fundamental
group
[cf.
Remarks
3.5.2,
(iii);
4.5.3,
(i)],
whereas
the
F
l
-
symmetry
arises
from
the
conjugation
action
of
the
absolute
Galois
group
of
the
global
base
field
[cf.
Remark
4.7.2]
—
i.e.,
where
we
recall
that
the
arithmetic
fundamental
groups
involved
may
be
thought
of
as
having
a
natural
hierarchical
structure
constituted
by
their
extension
structure
[corresponding
to
the
natural
outer
action
of
the
absolute
Galois
group
of
the
base
field
on
the
geometric
fundamental
group].
Remark
4.7.6.
One
important
observation
in
the
context
of
Corollary
4.7,
(i),
is
that
it
makes
sense
to
consider
non-realified
global
Frobenioids
[correspond-
ing,
e.g.,
to
“F
mod
”]
only
in
the
case
of
the
F
l
-symmetry.
Indeed,
in
order
to
consider
the
field
“F
mod
”
from
an
anabelian,
or
Galois-theoretic,
point
of
view,
it
is
necessary
to
consider
the
full
profinite
group
Π
C
F
—
i.e.,
not
just
the
open
sub-
groups
Π
C
K
,
Π
X
K
of
Π
C
F
which
give
rise,
respectively,
to
the
global
portions
of
the
±
F
l
-
and
F
l
-symmetries
[cf.
[IUTchI],
Definition
4.1,
(v);
[IUTchI],
Definition
6.1,
(v)].
On
the
other
hand,
to
work
with
the
abstract
topological
group
Π
C
F
means
that
the
subgroups
Π
C
K
,
Π
X
K
of
Π
C
F
are
only
well-defined
up
to
Π
C
F
-conjugacy.
That
is
to
say,
in
this
context,
the
subgroups
Π
C
K
,
Π
X
K
are
only
well-defined
up
to
automorphisms
arising
from
their
normalizers
in
Π
C
F
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.6,
(iii),
(iv)].
In
particular,
in
the
present
context,
one
is
obliged
to
regard
these
groups
Π
C
K
,
Π
X
K
as
being
subject
to
indeterminacies
arising
from
the
natural
F
l
-poly-actions
[i.e.,
actions
by
a
group
that
surjects
nat-
urally
onto
F
l
—
cf.
[IUTchI],
Example
4.3,
(iv)]
on
these
groups
—
that
is
to
say,
subject
to
indeterminacies
arising
from
the
natural
F
l
-symmetries
of
these
groups.
Here,
it
is
important
to
note
that
one
cannot
simply
“form
the
quotient
by
the
indeterminacy
constituted
by
these
F
l
-symmetries”
since
this
would
give
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
149
rise
to
“label-crushing”,
i.e.,
to
identifying
to
a
single
point
the
distinct
labels
j
∈
F
l
,
which
play
a
crucial
role
in
the
construction
of
the
Gaussian
monoids
[cf.
the
discussion
of
[IUTchI],
Remark
4.7.1].
But
then
the
F
l
-symmetries
of
Π
C
K
,
Π
X
K
that
one
must
contend
with
necessarily
involve
conjugation
by
elements
of
the
absolute
Galois
groups
of
the
global
base
fields
involved,
hence
are
fundamentally
incompatible
with
the
establishment
of
conjugate
synchronization
[cf.
the
discussion
of
Remark
4.7.2].
That
is
to
say,
just
as
it
is
necessary
to
(a)
isolate
the
F
±
l
-symmetry
from
the
F
l
-symmetry
in
order
to
establish
conjugate
synchronization
[cf.
the
discussion
of
Remark
4.7.2],
it
is
also
necessary
to
±
(b)
isolate
the
F
l
-symmetry
from
the
F
l
-symmetry
in
order
to
work
with
Galois-theoretic
representations
of
the
global
base
field
F
mod
.
Indeed,
in
this
context,
it
is
useful
to
recall
that
one
of
the
fundamental
themes
of
the
theory
of
the
present
series
of
papers
consists
precisely
of
the
dismantling
of
the
two
[a
priori
intertwined!]
combinatorial
dimensions
of
a
ring
[cf.
Remarks
4.7.3,
4.7.5;
[AbsTopIII],
§I3].
Remark
4.7.7.
The
theory
of
“tempered
versus
profinite
conjugates”
developed
in
[IUTchI],
§2,
is
applied
in
the
proof
of
Corollary
2.4,
(i),
in
a
setting
which
ultimately
[cf.
Remark
2.6.2,
(i);
Corollary
4.5,
(iii)]
is
seen
to
amount
to
a
certain
local
portion
[at
v
∈
V
bad
]
of
a
[D-]Θ
±ell
-Hodge
theater
—
i.e.,
a
setting
in
which
one
considers
the
F
±
l
-symmetry.
On
the
other
hand,
in
[IUTchI],
Remark
4.5.1,
(iii),
a
discussion
is
given
in
which
this
theory
of
“tempered
versus
profinite
conjugates”
developed
in
[IUTchI],
§2,
is
applied
in
a
setting
which
constitutes
a
certain
local
portion
[at
v
∈
V
bad
]
of
a
[D-]ΘNF-Hodge
theater.
In
this
context,
it
is
useful
to
note
that
the
point
of
view
of
this
discussion
given
in
[IUTchI],
Remark
4.5.1,
(iii),
may
be
regarded
as
“implicit”
in
the
point
of
view
of
the
theory
of
the
present
§4
in
the
following
sense:
The
profinite
conjugacy
indeterminacies
that
occur
in
an
[D-]ΘNF-Hodge
theater
[cf.
[IUTchI],
Remark
4.5.1,
(iii)]
are
linked
via
the
gluing
operation
discussed
in
[IUTchI],
Remark
6.12.2,
(i),
(ii)
—
cf.
Corollaries
4.6,
(iv);
4.7
—
to
the
profinite
conjugacy
indeterminacies
that
occur
in
an
[D-]Θ
±ell
-Hodge
theater
[cf.
Remarks
2.5.2,
(ii),
(iii);
2.6.2,
(i);
4.5.3,
(iii)],
i.e.,
to
the
profinite
conjugacy
indeterminacies
that
are
“resolved”
in
the
proof
of
Corollary
2.4,
(i),
by
applying
the
theory
of
[IUTchI],
§2.
Corollary
4.8.
(Frobenioid-theoretic
Monoids
Associated
to
ΘNF-
Hodge
Theaters)
Let
†
HT
ΘNF
=
(
†
F
†
F
†
NF
ψ
←−
†
F
J
†
Θ
ψ
−→
†
F
>
†
HT
Θ
)
be
a
ΘNF-Hodge
theater
[cf.
[IUTchI],
Definition
5.5,
(iii)]
which
lifts
the
D-
ΘNF-Hodge
theater
†
HT
D-ΘNF
of
Corollary
4.7
and
is
glued
to
the
Θ
±ell
-Hodge
±ell
of
Corollary
4.6
via
the
functorial
algorithm
of
[IUTchI],
Propo-
theater
†
HT
Θ
sition
6.7
[so
J
=
T
]
—
cf.
the
discussion
of
[IUTchI],
Remark
6.12.2,
(i),
(ii).
150
SHINICHI
MOCHIZUKI
(i)
(Non-realified
Global
Structures)
There
is
a
functorial
algorithm
in
the
category
†
F
[or
in
the
category
†
F
]
—
cf.
the
discussion
of
[IUTchI],
Ex-
ample
5.1,
(v),
(vi),
concerning
isomorphisms
of
cyclotomes
and
related
Kum-
mer
maps
—
for
constructing
Kummer
isomorphisms
of
pseudo-monoids
[the
first
two
of
which
are
equipped
with
group
actions
and
well-defined
up
to
a
single
conjugacy
indeterminacy]
∼
∼
κ-sol
†
†
κ-sol
†
†
†
π
1
(
D
)
M
∞
κ
→
π
1
(
D
)
M
∞
κ
(
D
)
,
†
M
κ
→
M
κ
(
D
)
and,
hence,
by
restricting
Kummer
classes
as
in
the
discussion
of
[IUTchI],
Example
5.1,
(v),
natural
“Kummer-theoretic”
isomorphisms
∼
π
1
(
†
D
)
†
M
→
π
1
(
†
D
)
M
(
†
D
)
π
1
(
†
D
)
†
M
π
1
κ-sol
(
†
D
)
†
†
M
sol
∼
→
∼
→
∼
†
M
mod
→
M
mod
(
D
),
π
1
(
†
D
)
M
(
†
D
)
π
1
κ-sol
(
†
D
)
†
†
M
sol
(
D
)
∼
M
mod
→
M
mod
(
†
D
)
—
which
may
be
interpreted
as
a
compatible
collection
of
isomorphisms
of
Frobenioids
†
†
∼
F
→
F
(
†
D
),
∼
F
mod
→
F
mod
(
†
D
),
∼
†
F
→
F
(
†
D
)
†
R
R
†
F
mod
→
F
mod
(
D
)
∼
[cf.
the
discussion
of
[IUTchI],
Example
5.1,
(ii),
(iii)].
(ii)
(Labels
and
F
l
-Symmetry)
In
the
notation
of
Corollary
4.7,
(ii),
the
collection
of
isomorphisms
of
Corollary
4.6,
(i)
[applied
to
the
F-prime-strips
of
the
capsule
†
F
J
;
cf.
also
the
discussion
of
[IUTchI],
Example
5.4,
(iv)],
together
∼
with
the
isomorphisms
of
(i)
above,
determine,
for
each
j
∈
LabCusp(
†
D
)
(
→
J)
[cf.
the
bijection
†
ζ
of
Corollary
4.7,
(ii)],
a
collection
of
isomorphisms
†
∼
F
j
→
†
∼
F
|
j
→
F
(
†
D
)|
j
∼
†
(
†
M
mod
)
j
→
M
mod
(
D
)
j
,
∼
(
†
M
mod
)
j
→
M
mod
(
†
D
)
j
∼
κ-sol
†
†
{π
1
κ-sol
(
†
D
)
†
M
(
D
)
M
sol
}
j
→
{π
1
sol
(
D
)}
j
∼
{π
1
κ-sol
(
†
D
)
†
M
}
→
{π
1
κ-sol
(
†
D
)
M
(
†
D
)}
j
∞
κ
j
∞
κ
∼
(
†
F
mod
)
j
→
F
mod
(
†
D
)
j
,
∼
R
R
†
(
†
F
mod
)
j
→
F
mod
(
D
)
j
as
well
as
[F
l
-]symmetrizing
isomorphisms,
induced
by
the
natural
poly-action
†
of
F
on
F
[cf.
[IUTchI],
Example
4.3,
(iv);
[IUTchI],
Corollary
5.3,
(i)],
be-
l
tween
the
data
indexed
by
distinct
j
∈
LabCusp(
†
D
).
Here,
[just
as
in
Corollary
4.7,
(ii)]
the
objects
equipped
with
π
1
rat
(
†
D
)(
π
1
κ-sol
(
†
D
))-actions
are
to
be
re-
rat/κ-sol
†
garded
as
being
subject
to
independent
π
1
(
D
)-conjugacy
indetermina-
rat
†
cies
for
distinct
j,
together
with
a
single
(π
1
(
D
)
)π
1
κ-sol
(
†
D
)-conjugacy
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
151
indeterminacy
that
is
independent
of
j
[cf.
the
discussion
of
the
final
portion
of
[IUTchI],
Example
5.1,
(i)].
Moreover,
these
symmetrizing
isomorphisms
are
com-
patible,
relative
to
†
ζ
[cf.
Corollary
4.7,
(ii)],
with
the
F
l
-symmetry
of
the
as-
sociated
NF-bridge
[cf.
[IUTchI],
Proposition
4.9,
(i);
[IUTchI],
Corollary
5.6,
(ii)]
and
determine
various
diagonal
F-prime-strips/submonoids/subrings/sub-
pseudo-monoids
[equipped
with
a
group
action
subject
to
conjugacy
indetermina-
cies
as
described
above]/subcategories
(−)
j
(−)
F
⊆
l
j∈F
l
[i.e.,
relative
to
the
conventions
discussed
in
Corollary
4.7,
(ii);
cf.
also
Remark
4.7.2].
(iii)
(Localization
Functors
and
Realified
Global
Structures)
Let
j
∈
LabCusp(
†
D
).
In
the
following,
objects
associated
to
an
F-prime-strip
labeled
by
j
at
an
element
v
∈
V
mod
will
be
denoted
by
means
of
a
label
“v
j
”.
Then
there
is
a
functorial
algorithm
in
the
NF-bridge
(
†
F
J
→
†
F
†
F
)
for
constructing
mutually
[1-]compatible
collections
of
“localization”
functors/poly-morphisms
[up
to
isomorphism]
)
j
(
†
F
mod
{π
1
κ-sol
(
†
D
)
†
→
†
R
(
†
F
mod
)
j
F
j
,
M
∞
κ
}
j
→
†
†
R
F
j
→
M
∞
κv
j
⊆
†
M
∞
κ×v
j
v∈V
as
in
the
discussion
of
[IUTchI],
Example
5.4,
(iv),
(vi)
[cf.
also
[IUTchI],
Defini-
tion
5.2,
(vi),
(viii)]
—
which
are
compatible,
relative
to
the
various
[Kummer/
“Kummer-theoretic”]
isomorphisms
of
(i),
(ii)
[cf.
also
[IUTchI],
Definition
5.2,
(vi),
(viii)],
with
the
collections
of
functors/poly-morphisms
of
Corollary
4.7,
(iii)
—
together
with
a
natural
isomorphism
of
Frobenioids
†
C
j
∼
→
R
(
†
F
mod
)
j
[cf.
the
notation
of
Corollary
4.6,
(ii);
[IUTchI],
Remark
5.2.1,
(ii),
applied
to
the
F-prime-strip
†
F
j
]
which
is
compatible
[cf.
Remark
4.8.3
below]
with
the
respective
bijections
involving
“Prime(−)”,
the
respective
local
isomorphisms
R
of
topological
monoids
[cf.
the
arrow
(
†
F
mod
)
j
→
†
F
R
j
discussed
above;
[IUTchI],
Remark
5.2.1,
(ii)],
the
isomorphisms
of
Corollary
4.7,
(iii),
and
the
vari-
ous
[“Kummer-theoretic”]
isomorphisms
of
(i),
(ii)
[cf.
also
Corollary
4.6,
(ii)].
Fi-
nally,
all
of
these
structures
are
compatible
with
the
respective
F
l
-symmetrizing
isomorphisms
[cf.
(ii)].
Proof.
The
various
assertions
of
Corollary
4.8
follow
immediately
from
the
defini-
tions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.8.1.
†
(i)
The
Frobenioid
C
gau
(
HT
Θ
)
of
Corollary
4.6,
(v),
is
constructed
as
a
sub-
†
†
category
of
a
product
over
j
∈
F
l
of
copies
C
j
of
the
category
C
.
In
particular,
152
SHINICHI
MOCHIZUKI
∼
R
one
may
apply
the
isomorphism
†
C
j
→
(
†
F
mod
)
j
of
Corollary
4.8,
(iii),
to
regard
Θ
†
this
Frobenioid
C
gau
(
HT
)
as
a
subcategory
†
(
HT
Θ
)
C
gau
→
R
(
†
F
mod
)
j
j∈F
l
†
R
of
the
product
over
j
∈
F
l
of
the
(
F
mod
)
j
.
(ii)
In
a
similar
vein,
the
local
data
at
v
∈
V
of
the
objects
Ψ
F
gau
(
†
HT
Θ
)
constructed
in
Corollary
4.6,
(iv),
gives
rise
to
[the
local
data
at
v
of
an
F
-prime-
strip,
i.e.,
in
particular,
to]
split
Frobenioids
F
gau
(
†
HT
Θ
)
v
[cf.
Definition
3.8,
(ii),
in
the
case
of
v
∈
V
bad
].
Write
F
gau
(
†
HT
Θ
)
for
the
F
-prime-strip
determined
by
this
local
data
F
gau
(
†
HT
Θ
)
v
at
v,
for
v
∈
V,
and
F
gau
(
†
HT
Θ
)
R
for
the
object
obtained
by
forming,
at
each
v
∈
V,
the
realification
of
the
underlying
Frobenioid
of
F
gau
(
†
HT
Θ
)
at
v.
Then
it
follows
from
the
construction
discussed
in
Corollary
4.6,
(iv),
that
one
may
think
of
the
realified
Frobenioid,
at
each
v
∈
V,
of
F
gau
(
†
HT
Θ
)
R
as
being
naturally
[“poly-”]embedded
F
gau
(
†
HT
Θ
)
R
→
(
†
F
R
>
)
j
j∈F
l
[where
we
use
this
notation
to
denote
the
collection
of
[“poly-”]embeddings
indexed
by
v
∈
V]
in
the
product
of
copies
of
realifications
of
[the
underlying
Frobenioids
of]
the
F-prime-strip
†
F
>
labeled
by
j
∈
F
l
.
Moreover,
by
applying
the
full
poly-
∼
†
†
isomorphisms
(
F
>
)
j
→
F
j
—
which
are
tautologically
compatible
with
the
labels
Θ
R
†
j
∈
F
l
!
—
we
may
think
of
F
gau
(
HT
)
as
being
naturally
[“poly-”]embedded
F
gau
(
†
HT
Θ
)
R
→
†
R
F
j
j∈F
l
[where
we
use
this
notation
to
denote
the
collection
of
[“poly-”]embeddings
in-
dexed
by
v
∈
V]
in
the
product
associated
to
the
realifications
of
[the
underlying
Frobenioids
of]
the
F-prime-strips
†
F
j
.
(iii)
Thus,
by
applying
the
various
[“poly-”]embeddings
considered
in
(i),
(ii),
one
may
think
of
the
“realified
localization”
functors
R
)
j
(
†
F
mod
→
†
R
F
j
of
Corollary
4.8,
(iii),
as
inducing
a
“realified
localization”
functor
[up
to
isomor-
phism]
†
(
HT
Θ
)
→
F
gau
(
†
HT
Θ
)
R
C
gau
—
which
[as
one
verifies
immediately]
is
compatible
[cf.
the
various
compatibil-
ities
discussed
in
Corollary
4.8,
(iii)]
with
the
realified
localization
isomorphisms
∼
Θ
R
†
)
v
,
for
v
∈
V,
considered
in
Corollary
4.6,
(v).
Φ
C
gau
(
†
HT
Θ
),v
→
Ψ
F
gau
(
HT
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
153
Remark
4.8.2.
(i)
The
realified
localization
functor
discussed
in
Remark
4.8.1,
(iii),
only
concerns
the
realification
of
the
Frobenioid-theoretic
version
F
gau
(
†
HT
Θ
)
of
the
Gaussian
monoids.
The
unit
portion
of
the
Gaussian
monoids
will
be
used,
in
the
context
of
the
theory
involving
the
log-wall
that
will
be
developed
in
[IUTchIII],
not
in
its
capacity
as
a
“multiplicative
object”,
but
rather
—
i.e.,
by
applying
the
operation
“log”
to
the
units
at
the
various
v
∈
V,
as
in
the
theory
of
[AbsTopIII]
—
as
an
“additive
object”.
In
this
theory,
the
non-realified
global
Frobenioids
of
Corollary
4.8,
(i),
will
appear
in
the
context
of
localization
functors/morphisms
—
i.e.,
as
a
sort
of
translation
apparatus
between
-
and
-line
bundles
[cf.
the
discussion
of
Remark
4.7.2]
—
that
relate
these
[multiplicative!]
non-realified
global
Frobenioids
to
the
[additive!]
images
via
“log”
of
the
units.
Note
that
this
sort
of
construction
—
i.e.,
in
which
the
localization
operations
involving
units
and
value
groups
differ
by
a
shift
via
the
operation
“log”
—
depends,
in
an
essential
way
[cf.
the
discussion
of
Remark
1.12.2,
(iv)],
on
the
natural
splittings
with
which
the
Gaussian
monoids
are
equipped
[cf.
Corollary
4.6,
(iv)].
(ii)
In
the
context
of
(i),
it
is
useful
to
observe
that,
although
the
non-realified
global
Frobenioids
of
Corollary
4.8,
(i),
may
only
be
considered
in
the
context
of
the
F
l
-symmetry
[cf.
the
discussion
of
Remark
4.7.6],
this
does
not
yield
any
obstacles,
relative
to
the
discussion
in
(i)
of
Gaussian
monoids,
since
Gaussian
monoids
are
most
naturally
considered
as
“functions”
of
a
parameter
j
∈
F
l
[cf.
the
discussion
of
Remark
4.7.3,
(iii)].
(iii)
From
the
point
of
view
of
the
analogy
of
the
theory
of
the
present
series
of
papers
with
p-adic
Teichmüller
theory
[cf.
the
discussion
of
[AbsTopIII],
§I5],
it
is
of
interest
to
note
that
the
construction
discussed
in
(i)
involving
the
use
of
the
natural
splittings
of
Gaussian
monoids
to
consider
“log-shifted
units”
together
with
“non-log-shifted
value
groups”
may
be
thought
of
as
corresponding
to
the
situation
that
frequently
occurs
in
p-adic
Teichmüller
theory
in
which
an
indigenous
bundle
(E,
∇
E
)
equipped
with
a
Hodge
filtration
0
→
ω
→
E
→
τ
→
0
on
a
hyperbolic
curve
in
positive
characteristic
is
represented,
in
the
context
of
local
Frobenius
liftings
modulo
higher
powers
of
p,
as
a
direct
sum
Φ
∗
τ
⊕
ω
—
where
Φ
denotes
the
Frobenius
morphism
on
the
curve,
which,
as
may
be
recalled
from
the
discussion
of
[AbsTopIII],
§I5,
corresponds,
relative
to
the
analogy
under
consideration,
to
the
operation
“log”
studied
in
[AbsTopIII].
Remark
4.8.3.
Similar
observations
to
the
observations
made
in
Remark
4.5.2,
(i),
(ii),
concerning
the
F
±
l
-symmetrizing
isomorphisms
of
Corollary
4.5,
(iii),
may
be
made
in
the
case
of
the
F
l
-symmetrizing
isomorphisms
of
Corollary
4.8,
(ii).
Definition
4.9.
(i)
Let
C
be
an
arbitrary
Frobenioid.
Write
D
for
the
base
category
of
C.
Suppose
that
D
is
isomorphic
to
the
category
of
connected
finite
étale
coverings
154
SHINICHI
MOCHIZUKI
of
the
spectrum
of
an
MLF
or
a
CAF.
Let
A
be
a
“universal
covering
pro-object”
def
of
D
[cf.
the
discussion
of
Example
3.2,
(i),
(ii)].
Write
G
=
Aut(A)
[so
G
is
isomorphic
to
the
absolute
Galois
group
of
an
MLF
or
a
CAF].
Now
by
evaluating
the
monoid
“O
(−)”
on
D
that
arises
from
the
general
theory
of
Frobenioids
[cf.
[FrdI],
Proposition
2.2]
at
A,
we
thus
obtain
a
monoid
[in
the
usual
sense]
equipped
with
a
natural
action
by
G
G
O
(A)
[cf.
the
discussion
of
Example
3.2,
(ii)].
If
N
is
a
positive
integer,
then
we
shall
write
μ
N
(A)
⊆
O
μ
(A)
⊆
O
×
(A)
for
the
subgroups
of
N
-torsion
elements
[cf.
[FrdII],
Definition
2.1,
(i)]
and
torsion
elements
of
arbitrary
order;
O
×
(A)
O
×μ
N
(A)
O
×μ
(A)
for
the
respective
quotients
of
the
submonoid
of
units
O
×
(A)
⊆
O
(A)
by
μ
N
(A),
O
μ
(A).
Thus,
O
(A),
O
×
(A),
O
×μ
N
(A),
O
×μ
(A),
μ
N
(A),
and
O
μ
(A)
are
all
equipped
with
natural
G-actions.
Next,
let
us
suppose
that
G
is
nontrivial
[i.e.,
arises
from
an
MLF].
Recall
the
group-theoretic
algorithms
“G
→
(G
O
×
(G))”
and
“G
→
(G
O
×μ
(G))”
discussed
in
Example
1.8,
(iii),
(iv).
We
define
a
×-Kummer
structure
(respectively,
×μ-Kummer
structure)
on
C
to
be
a
Z
×
-
(respectively,
Ism-
[cf.
Example
1.8,
(iv)])
orbit
of
isomorphisms
κ
×
:
O
×
(G)
∼
→
O
×
(A)
(respectively,
κ
×μ
:
O
×μ
(G)
∼
→
O
×μ
(A))
of
ind-topological
G-modules.
Note
that
since
any
two
“universal
covering
pro-
objects”
of
D
are
isomorphic,
it
follows
immediately
that
the
definition
of
a
×-
(respectively,
×μ-)
Kummer
structure
is
independent
of
the
choice
of
A.
Next,
let
us
recall
from
Remark
1.11.1,
(b),
that
any
×-Kummer
structure
on
C
is
unique.
In
the
case
of
×μ-Kummer
structures,
let
us
observe
that
a
×μ-Kummer
structure
κ
×μ
on
C
determines,
for
each
open
subgroup
H
⊆
G,
a
submodule
κ
(A)
I
H
def
=
Im(O
×
(G)
H
)
⊆
O
×μ
(A)
—
namely,
the
image
via
κ
×μ
of
the
image
of
O
×
(G)
H
in
O
×μ
(G)
H
[where
the
superscript
“H’s”
denote
the
submodules
of
H-invariants].
Conversely,
it
is
es-
sentially
a
tautology
[cf.
the
definition
of
“Ism”
given
in
Example
1.8,
(iv)!]
that
the
×μ-Kummer
structure
κ
×μ
on
C
is
completely
determined
by
the
submodules
κ
(A)
⊆
O
×μ
(A)}
H
[where
H
ranges
over
the
open
subgroups
of
G],
namely,
as
{I
H
∼
the
unique
Ism-orbit
of
G-equivariant
isomorphisms
O
×μ
(G)
→
O
×μ
(A)
that
maps
κ
(A)
for
each
open
subgroup
H
⊆
G.
That
is
to
say,
O
×
(G)
H
onto
I
H
a
×μ-Kummer
structure
κ
×μ
on
C
may
be
thought
of
as
—
i.e.,
in
the
sense
that
it
determines
and
is
uniquely
determined
by
—
the
collection
κ
(A)
⊆
O
×μ
(A)}
H
[where
H
ranges
over
the
open
of
submodules
{I
H
subgroups
of
G].
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
155
Finally,
we
shall
refer
to
as
a
[×-,
×μ-]Kummer
Frobenioid
any
Frobenioid
equipped
with
a
[×-,
×μ-]Kummer
structure.
We
shall
refer
to
as
a
split-[×-,
×μ-]Kummer
Frobenioid
any
split
Frobenioid
equipped
with
a
[×-,
×μ-]Kummer
structure.
(ii)
Let
‡
F
=
{
‡
F
v
}
v∈V
be
an
F
-prime-strip;
w
∈
V
bad
.
Write
‡
D
=
{
‡
D
v
}
v∈V
for
the
D
-prime-
strip
associated
to
‡
F
[cf.
[IUTchI],
Remark
5.2.1,
(i)].
Thus,
‡
F
w
is
a
split
Frobenioid
[cf.
[IUTchI],
Definition
5.2,
(ii),
(a);
[IUTchI],
Example
3.2,
(v)],
with
.
Let
‡
A
be
a
“universal
covering
pro-object”
of
‡
D
w
[cf.
the
base
category
‡
D
w
def
discussion
of
(i)].
Write
‡
G
=
Aut(
‡
A)
[so
‡
G
is
a
profinite
group
isomorphic
to
G
w
].
Then
the
2l-torsion
subgroup
μ
2l
(
‡
A)
⊆
O
×
(
‡
A)
of
the
submonoid
of
units
O
×
(
‡
A)
⊆
O
(
‡
A)
of
O
(
‡
A),
together
with
the
images
of
the
splittings
with
which
‡
F
w
is
equipped,
generate
a
submonoid
O
⊥
(
‡
A)
⊆
O
(
‡
A),
whose
quotient
by
μ
2l
(
‡
A)
we
denote
by
O
(
‡
A)
⊇
O
⊥
(
‡
A)
O
(
‡
A)
O
⊥
(
‡
A)/μ
2l
(
‡
A)
def
=
∼
[so
we
have
a
natural
isomorphism
O
(
‡
A)/O
×
(
‡
A)
→
O
(
‡
A)].
Write
O
×μ
(
‡
A)
O
(
‡
A)
×
O
×μ
(
‡
A)
def
=
for
the
direct
product
monoid.
Thus,
the
monoids
O
(
‡
A),
O
⊥
(
‡
A),
O
(
‡
A),
O
×
(
‡
A),
O
×μ
(
‡
A),
O
μ
(
‡
A),
and
O
×μ
(
‡
A)
are
all
equipped
with
natural
‡
G-
actions.
Next,
we
consider
the
group-theoretic
algorithms
“G
→
(G
O
×
(G))”
and
“G
→
(G
O
×μ
(G))”
discussed
in
Example
1.8,
(iii),
(iv).
If
we
apply
the
first
of
these
algorithms
to
‡
G,
then
it
follows
from
Remark
1.11.1,
(b),
that
there
exists
a
unique
Z
×
-orbit
of
isomorphisms
‡
×
κ
w
:
O
×
(
‡
G)
∼
→
O
×
(
‡
A)
of
ind-topological
modules
equipped
with
‡
G-actions.
Moreover,
‡
κ
×
w
induces
an
Ism-orbit
[cf.
Example
1.8,
(iv)]
of
isomorphisms
‡
×μ
κ
w
:
O
×μ
(
‡
G)
∼
→
O
×μ
(
‡
A)
—
i.e.,
by
forming
the
quotient
by
“O
μ
(−)”.
(iii)
In
the
notation
of
(ii),
the
[rational
function
monoid
determined
by
the
groupification
of
the]
monoid
with
‡
G-action
O
×μ
(
‡
A),
together
with
the
divisor
monoid
of
[the
underlying
Frobenioid
of]
‡
F
w
,
determines
a
“model
Frobenioid”
[cf.
[FrdI],
Theorem
5.2,
(ii)]
equipped
with
a
splitting,
i.e.,
the
splitting
arising
from
the
definition
of
O
×μ
(
‡
A)
as
a
direct
product.
Thus,
the
‡
G-module
obtained
by
evaluating
at
‡
A
the
group
of
units
“O
×
(−)”
(respectively,
the
monoid
“O
(−)”)
associated
to
this
Frobenioid
may
be
naturally
identified
with
O
×μ
(
‡
A)
(respec-
determines
tively,
O
×μ
(
‡
A)).
In
particular,
the
Ism-orbit
of
isomorphisms
‡
κ
×μ
w
a
×μ-Kummer
structure
on
this
Frobenioid.
We
shall
write
‡
F
w
×μ
156
SHINICHI
MOCHIZUKI
for
the
resulting
split-Kummer
Frobenioid
and
—
by
abuse
of
notation!
—
‡
F
w
for
the
split-Kummer
Frobenioid
determined
by
the
split
Frobenioid
‡
F
w
equipped
with
the
×-Kummer
structure
determined
by
‡
κ
×
w
.
Here,
we
remark
that
the
primary
justification
for
this
abuse
of
notation
lies
in
the
uniqueness
of
×-Kummer
structures
discussed
in
(i)
above.
(iv)
Let
‡
F
be
as
in
(ii);
w
∈
V
good
V
non
.
Thus,
‡
F
w
is
a
split
Frobenioid
[cf.
[IUTchI],
Definition
5.2,
(ii),
(a);
[IUTchI],
Example
3.3,
(i)],
with
base
category
‡
D
w
.
Let
‡
A
be
a
“universal
covering
pro-object”
of
‡
D
w
[cf.
the
discussion
of
def
(i)].
Write
‡
G
=
Aut(
‡
A)
[so
‡
G
is
a
profinite
group
isomorphic
to
G
w
].
Then
the
image
of
the
splitting
with
which
‡
F
w
is
equipped
determines
a
submonoid
def
O
⊥
(
‡
A)
⊆
O
(
‡
A).
Write
O
(
‡
A)
=
O
⊥
(
‡
A),
O
×μ
(
‡
A)
O
(
‡
A)
×
O
×μ
(
‡
A)
def
=
for
the
direct
product
monoid.
Thus,
the
monoids
O
(
‡
A),
O
⊥
(
‡
A),
O
(
‡
A),
O
×
(
‡
A),
O
×μ
(
‡
A),
O
μ
(
‡
A),
and
O
×μ
(
‡
A)
are
all
equipped
with
natural
‡
G-
actions.
Next,
we
consider
the
group-theoretic
algorithms
“G
→
(G
O
×
(G))”
and
“G
→
(G
O
×μ
(G))”
discussed
in
Example
1.8,
(iii),
(iv).
If
we
apply
the
first
of
these
algorithms
to
‡
G,
then
it
follows
from
Remark
1.11.1,
(b),
that
there
exists
a
unique
Z
×
-orbit
of
isomorphisms
‡
×
κ
w
:
O
×
(
‡
G)
∼
→
O
×
(
‡
A)
of
ind-topological
modules
equipped
with
‡
G-actions.
Moreover,
‡
κ
×
w
induces
an
Ism-orbit
[cf.
Example
1.8,
(iv)]
of
isomorphisms
‡
×μ
κ
w
:
O
×μ
(
‡
G)
∼
→
O
×μ
(
‡
A)
—
i.e.,
by
forming
the
quotient
by
“O
μ
(−)”.
The
[rational
function
monoid
de-
termined
by
the
groupification
of
the]
monoid
with
‡
G-action
O
×μ
(
‡
A),
together
with
the
divisor
monoid
of
[the
underlying
Frobenioid
of]
‡
F
w
,
determines
a
“model
Frobenioid”
[cf.
[FrdI],
Theorem
5.2,
(ii)]
equipped
with
a
splitting,
i.e.,
the
splitting
arising
from
the
definition
of
O
×μ
(
‡
A)
as
a
direct
product.
Thus,
the
‡
G-module
obtained
by
evaluating
at
‡
A
the
group
of
units
“O
×
(−)”
(respectively,
the
monoid
“O
(−)”)
associated
to
this
Frobenioid
may
be
naturally
identified
with
O
×μ
(
‡
A)
de-
(respectively,
O
×μ
(
‡
A)).
In
particular,
the
Ism-orbit
of
isomorphisms
‡
κ
×μ
w
termines
a
×μ-Kummer
structure
on
this
Frobenioid.
We
shall
write
‡
F
w
×μ
for
the
resulting
split-Kummer
Frobenioid
and
—
by
abuse
of
notation!
[cf.
the
discussion
of
(iii)
above]
—
‡
F
w
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
157
for
the
split-Kummer
Frobenioid
determined
by
the
split
Frobenioid
‡
F
w
equipped
with
the
×-Kummer
structure
determined
by
‡
κ
×
w
.
(v)
Let
‡
F
be
as
in
(ii);
w
∈
V
arc
.
Then
we
shall
write
‡
F
w
×μ
for
the
collection
of
data
obtained
by
replacing
the
split
Frobenioid
that
appears
in
the
collection
of
data
‡
F
w
[cf.
[IUTchI],
Definition
5.2,
(ii),
(b);
[IUTchI],
Example
3.4,
(ii)]
by
the
inductive
system,
indexed
by
any
[“multiplicatively”]
cofinal
subset
of
the
multiplicative
monoid
N
≥1
,
of
split
Frobenioids
obtained
[in
the
evident
fashion]
from
‡
F
w
by
forming
the
quotients
by
the
N
-torsion,
for
N
∈
N
≥1
.
Here,
we
identify
[in
the
evident
fashion]
the
inductive
systems
arising
from
distinct
cofinal
subsets
of
N
≥1
.
Thus,
[cf.
the
notation
of
(i)]
the
units
of
the
split
Frobenioids
of
this
inductive
system
give
rise
to
an
inductive
system
...
O
×μ
N
(A)
...
O
×μ
N
·N
(A)
...
is
an
object
of
the
category
TM
[cf.
[where
N,
N
∈
N
≥1
].
Now
recall
that
‡
D
w
×
[IUTchI],
Definition
4.1,
(iii),
(b)].
In
particular,
the
units
(
‡
D
w
)
of
this
object
of
1
TM
form
a
topological
group
[noncanonically
isomorphic
to
S
],
which
we
think
of
as
being
related
to
the
above
inductive
system
of
units
via
a
system
of
compatible
surjections
×
)
O
×μ
N
(A)
(
‡
D
w
[i.e.,
where
the
kernel
of
the
displayed
surjection
is
the
subgroup
of
N
-torsion].
This
system
of
compatible
surjections
is
well-defined
up
to
an
indeterminacy
given
by
×
)
.
When
considered
composition
with
the
unique
nontrivial
automorphism
of
(
‡
D
w
up
to
this
indeterminacy,
this
system
of
compatible
surjections
may
be
thought
of
as
a
sort
of
Kummer
structure
on
‡
F
w
×μ
[which
may
be
algorithmically
reconstructed
from
the
collection
of
data
‡
F
w
×μ
].
(vi)
Write
‡
×μ
F
=
{
‡
F
v
×μ
}
v∈V
for
the
collection
of
data
indexed
by
V
obtained
as
follows:
(a)
if
v
∈
V
bad
,
then
we
take
‡
F
v
×μ
to
be
the
split-Kummer
Frobenioid
constructed
in
(iii);
(b)
if
v
∈
V
good
V
non
,
then
we
take
‡
F
v
×μ
to
be
the
split-Kummer
Frobenioid
constructed
in
(iv);
(c)
if
v
∈
V
arc
,
then
we
take
‡
F
v
×μ
to
be
the
collection
of
data
constructed
in
(v).
Moreover,
by
replacing
the
various
split
Frobenioids
of
‡
F
(respectively,
‡
×μ
F
)
with
the
split
Frobenioids
—
i.e.,
equipped
with
trivial
splittings!
—
obtained
by
considering
the
subcategories
[of
the
underlying
categories
associated
to
these
Frobenioids]
determined
by
the
isometries
[i.e.,
roughly
speaking,
the
“units”
—
cf.
[FrdI],
Theorem
5.1,
(iii),
in
the
case
of
v
∈
V
non
;
[FrdII],
Example
3.3,
(iii),
in
the
case
of
v
∈
V
arc
],
one
obtains
a
collection
of
data
‡
×
F
=
{
‡
F
v
×
}
v∈V
(respectively,
‡
F
×μ
=
{
‡
F
v
×μ
}
v∈V
)
indexed
by
V.
Thus,
for
each
v
∈
V
non
,
‡
F
v
×
(respectively,
‡
F
v
×μ
)
is
a
split-×-
(respectively,
split-×μ-)
Kummer
Frobenioid.
158
SHINICHI
MOCHIZUKI
(vii)
Let
∈
{
×,
×μ,
×μ
}.
Then
we
define
an
F
-prime-strip
to
be
a
collection
of
data
∗
F
=
{
∗
F
v
}
v∈V
such
that
for
each
v
∈
V,
∗
F
v
is
a
collection
of
data
that
is
isomorphic
to
‡
F
v
[cf.
(vi)].
A
morphism
of
F
-prime-strips
is
defined
to
be
a
collection
of
isomor-
phisms,
indexed
by
V,
between
the
various
constituent
objects
of
the
prime-strips
[cf.
[IUTchI],
Definition
5.2,
(iii)].
(viii)
We
define
an
F
×μ
-prime-strip
to
be
a
collection
of
data
∗
×μ
F
∼
=
(
∗
C
,
Prime(
∗
C
)
→
V,
∗
F
×μ
,
{
∗
ρ
v
}
v∈V
)
satisfying
the
conditions
(a),
(b),
(c),
(d),
(e),
(f)
of
[IUTchI],
Definition
5.2,
(iv),
for
an
F
-prime-strip,
where
the
portion
of
the
collection
of
data
constituted
by
an
F
-prime-strip
is
replaced
by
an
F
×μ
-prime-strip.
Thus,
relative
to
the
notation
of
the
above
display
[cf.
also
(ii),
(iii)],
the
generators
of
the
monoids
“O
(−)”
[each
of
which
is
abstractly
isomorphic
to
N]
of
the
data
at
v
∈
V
bad
(
=
∅)
[cf.
[IUTchI],
Definition
3.1,
(b)]
of
∗
F
×μ
=
{
∗
F
w
×μ
}
w∈V
,
together
with
the
{
∗
ρ
w
}
w∈V
,
determine
a
well-defined
object,
up
to
isomorphism,
of
the
global
realified
Frobenioid
∗
C
of
negative
“arithmetic
degree”
[cf.
[FrdI],
Example
6.3;
[FrdI],
Theorem
6.4,
(i),
(ii)],
which
we
refer
to
as
the
pilot
object
associated
to
the
F
×μ
-prime-strip
∗
F
×μ
.
A
morphism
of
F
×μ
-prime-strips
is
defined
to
be
an
isomorphism
between
collections
of
data
as
discussed
above.
We
conclude
the
present
paper
with
the
following
two
results,
which
may
be
thought
of
as
enhanced
versions
of
[IUTchI],
Corollaries
3.7,
3.8,
3.9
—
i.e.,
versions
that
reflect
the
various
enhancements
made
to
the
theory
in
[IUTchI],
§4,
§5,
§6,
as
well
as
in
the
present
paper.
Corollary
4.10.
(Frobenius-pictures
of
Θ
±ell
NF-Hodge
Theaters)
Fix
a
collection
of
initial
Θ-data
(F
/F,
X
F
,
l,
C
K
,
V,
V
bad
mod
,
)
as
in
[IUTchI],
Definition
3.1.
Let
±ell
±ell
†
HT
Θ
NF
;
‡
HT
Θ
NF
be
Θ
±ell
NF-Hodge
theaters
[relative
to
the
given
initial
Θ-data]
—
cf.
[IUTchI],
±ell
±ell
Definition
6.13,
(i).
Write
†
HT
D-Θ
NF
,
‡
HT
D-Θ
NF
for
the
associated
D-
Θ
±ell
NF-Hodge
theaters
—
cf.
[IUTchI],
Definition
6.13,
(ii).
Then:
(i)
(Constant
Prime-Strips)
Let
us
apply
the
constructions
of
Corollary
±ell
4.6,
(i),
(iii),
to
the
underlying
Θ
±ell
-Hodge
theater
of
†
HT
Θ
NF
.
Then,
for
each
t
∈
LabCusp
±
(
†
D
),
the
collection
of
data
Ψ
cns
(
†
F
)
t
determines,
in
a
natural
way,
an
F-prime-strip
[cf.
Remark
4.6.2,
(i)].
Let
us
identify
the
collections
of
data
Ψ
cns
(
†
F
)
0
and
Ψ
cns
(
†
F
)
F
l
via
the
isomorphism
of
the
second
display
of
Corollary
4.6,
(iii),
and
denote
by
†
F
∼
=
(
†
C
,
Prime(
†
C
)
→
V,
†
F
,
{
†
ρ
,v
}
v∈V
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
159
the
resulting
F
-prime-strip
determined
by
the
constructions
discussed
in
[IUTchI],
Remark
5.2.1,
(ii)
[which,
as
is
easily
verified,
are
compatible
with
the
F
±
l
-
symmetrizing
isomorphisms
of
Corollary
4.6,
(iii)].
Thus,
[it
follows
imme-
diately
from
the
constructions
involved
that]
one
has
a
natural
identification
∼
†
and
the
collec-
isomorphism
of
F
-prime-strips
†
F
→
†
F
mod
between
F
±ell
Θ
†
tion
of
data
†
F
mod
associated
to
the
underlying
Θ-Hodge
theater
of
HT
[IUTchI],
Definition
3.6,
(c)]
—
cf.
the
discussion
of
the
assignment
“
0,
→
NF
[cf.
>
”
in
Remark
3.8.2,
(ii).
(ii)
(Theta
and
Gaussian
Prime-Strips)
Let
us
apply
the
constructions
of
Corollary
4.6,
(iv),
(v),
to
the
underlying
Θ-bridge
and
Θ
±ell
-Hodge
theater
of
±ell
†
HT
Θ
NF
.
Then
the
collection
of
data
Ψ
F
env
(
†
HT
Θ
)
[cf.
Corollary
4.6,
(iv)],
def
†
=
C
env
(
HT
Θ
)
[cf.
Corollary
4.6,
(v)],
and
the
the
global
realified
Frobenioid
†
C
env
∼
local
isomorphisms
Φ
C
env
→
Ψ
F
env
(
†
HT
Θ
)
R
(
†
HT
Θ
),v
v
for
v
∈
V
[cf.
Corollary
4.6,
(v)]
give
rise,
in
a
natural
fashion,
to
an
F
-prime-strip
†
F
env
∼
=
(
†
C
env
,
Prime(
†
C
env
)
→
V,
†
F
env
,
{
†
ρ
env,v
}
v∈V
)
[so,
in
particular,
†
F
env
is
the
F
-prime-strip
determined
by
Ψ
F
env
(
†
HT
Θ
)
—
cf.
Remark
4.6.2,
(i);
Remark
4.10.1
below].
Thus,
[it
follows
immediately
from
the
constructions
involved
that]
there
is
a
natural
identification
isomorphism
of
∼
†
†
F
tht
between
†
F
F
-prime-strips
†
F
env
→
env
and
the
collection
of
data
F
tht
as-
±ell
sociated
to
the
underlying
Θ-Hodge
theater
of
†
HT
Θ
NF
[cf.
[IUTchI],
Definition
3.6,
(c)].
In
a
similar
vein,
the
collection
of
data
Ψ
F
gau
(
†
HT
Θ
)
[cf.
Corollary
def
†
4.6,
(iv)],
the
global
realified
Frobenioid
†
C
gau
=
C
gau
(
HT
Θ
)
[cf.
Corollary
4.6,
∼
(v)],
and
the
local
isomorphisms
Φ
C
gau
→
Ψ
F
gau
(
†
HT
Θ
)
R
(
†
HT
Θ
),v
v
for
v
∈
V
[cf.
Corollary
4.6,
(v)]
give
rise,
in
a
natural
fashion,
to
an
F
-prime-strip
†
F
gau
∼
=
(
†
C
gau
,
Prime(
†
C
gau
)
→
V,
†
F
gau
,
{
†
ρ
gau,v
}
v∈V
)
[so,
in
particular,
†
F
gau
is
the
F
-prime-strip
determined
by
Ψ
F
gau
(
†
HT
Θ
)
—
cf.
Remark
4.6.2,
(i);
Remark
4.10.1
below].
Finally,
the
evaluation
isomorphisms
of
Corollary
4.6,
(iv),
(v),
determine
an
evaluation
isomorphism
†
F
env
∼
→
†
F
gau
of
F
-prime-strips.
‡
×μ
(respectively,
†
F
×μ
;
†
F
×μ
)
(iii)
(Θ
×μ
-
and
Θ
×μ
gau
-Links)
Write
F
env
gau
×μ
‡
†
-prime-strip
associated
to
the
F
-prime-strip
F
(respectively,
F
env
;
for
the
F
†
F
gau
)
[cf.
Definition
4.9,
(viii);
the
functorial
algorithm
described
in
Definition
4.9,
(vi)].
Then
the
functoriality
of
this
algorithm
induces
maps
‡
Isom
F
(
†
F
env
,
F
)
→
Isom
F
×μ
(
†
F
×μ
,
‡
F
×μ
)
env
‡
Isom
F
(
†
F
gau
,
F
)
→
Isom
F
×μ
(
†
F
×μ
,
‡
F
×μ
)
gau
160
SHINICHI
MOCHIZUKI
from
[nonempty!]
sets
of
isomorphisms
of
F
-prime-strips
to
[nonempty!]
sets
of
isomorphisms
of
F
×μ
-prime-strips.
Here,
the
second
map
may
be
regarded
as
being
obtained
from
the
first
map
via
composition
[in
the
case
of
the
domain
∼
†
“Isom
F
(−,
−)”]
with
the
evaluation
isomorphism
†
F
F
gau
of
(ii)
and
env
→
composition
[in
the
case
of
the
codomain
“Isom
F
×μ
(−,
−)”]
with
the
isomorphism
†
×μ
∼
†
×μ
F
env
→
F
gau
functorially
obtained
from
this
isomorphism
of
(ii).
We
shall
∼
refer
to
the
full
poly-isomorphism
†
F
×μ
→
‡
F
×μ
as
the
Θ
×μ
-link
env
†
HT
Θ
±ell
NF
Θ
×μ
−→
‡
HT
Θ
±ell
NF
±ell
[cf.
the
“Θ-link”
of
[IUTchI],
Corollary
3.7,
(i)]
from
†
HT
Θ
NF
to
‡
HT
Θ
∼
and
to
the
full
poly-isomorphism
†
F
×μ
→
‡
F
×μ
as
the
Θ
×μ
gau
gau
-link
†
from
†
HT
Θ
±ell
NF
HT
Θ
±ell
to
‡
HT
Θ
±ell
NF
NF
Θ
×μ
gau
−→
‡
±ell
HT
Θ
±ell
NF
,
NF
.
(iv)
(Coric
F
×μ
-Prime-Strips)
The
definition
of
the
unit
portion
of
the
theta
and
Gaussian
monoids
involved
[cf.
Corollary
3.5,
(ii);
Corollary
3.6,
(ii);
Proposition
4.1,
(iv);
Proposition
4.2,
(iv);
Proposition
4.3,
(iv);
Proposition
4.4,
(iv)]
gives
rise
to
natural
isomorphisms
†
×μ
F
∼
→
∼
†
×μ
F
env
→
†
×μ
F
gau
†
×μ
×μ
—
where
we
write
†
F
×μ
,
†
F
×μ
-prime-strips
associated
to
env
,
F
gau
for
the
F
†
†
†
the
F
-prime-strips
F
,
F
env
,
F
gau
,
respectively
[cf.
the
functorial
algorithm
described
in
Definition
4.9,
(vi)].
Moreover,
by
composing
these
natural
isomor-
phisms
with
the
poly-isomorphisms
induced
on
the
respective
F
×μ
-prime-strips
by
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
(iii),
one
obtains
a
poly-isomorphism
∼
†
×μ
F
→
‡
×μ
F
which
coincides
with
the
full
poly-isomorphism
between
these
two
F
×μ
-prime-
strips
—
that
is
to
say,
“
(−)
F
×μ
”
is
an
invariant
of
both
the
Θ
×μ
-
and
Θ
×μ
gau
-links.
Finally,
this
full
poly-isomorphism
induces
[cf.
Definition
4.9,
(vii);
[IUTchI],
Remark
5.2.1,
(i)]
the
full
poly-isomorphism
†
∼
D
→
‡
D
between
the
associated
D
-prime-strips;
we
shall
refer
to
this
poly-isomorphism
as
±ell
±ell
the
D-Θ
±ell
NF-link
from
†
HT
D-Θ
NF
to
‡
HT
D-Θ
NF
.
(v)
(Coric
Global
Realified
Frobenioids)
The
full
poly-isomorphism
†
D
→
‡
D
of
(iv)
induces
[cf.
Corollary
4.5,
(ii)]
an
isomorphism
of
collections
of
data
∼
∼
(D
(
†
D
),
Prime(D
(
†
D
))
→
V,
{
†
ρ
D
,v
}
v∈V
)
∼
→
∼
(D
(
‡
D
),
Prime(D
(
‡
D
))
→
V,
{
‡
ρ
D
,v
}
v∈V
)
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
161
—
i.e.,
consisting
of
a
Frobenioid,
a
bijection,
and
a
collection
of
isomorphisms
of
topological
monoids
indexed
by
V.
Moreover,
this
isomorphism
of
collections
of
data
is
compatible,
relative
to
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
(iii),
with
the
R
>0
-orbits
of
the
isomorphisms
of
collections
of
data
∼
(
†
C
,
Prime(
†
C
)
→
V,
{
†
ρ
∼
,v
}
v∈V
)
∼
(D
(
†
D
),
Prime(D
(
†
D
))
→
V,
{
†
ρ
D
,v
}
v∈V
)
→
∼
(
‡
C
,
Prime(
‡
C
)
→
V,
{
‡
ρ
∼
,v
}
v∈V
)
∼
(D
(
‡
D
),
Prime(D
(
‡
D
))
→
V,
{
‡
ρ
D
,v
}
v∈V
)
→
obtained
by
applying
the
functorial
algorithm
discussed
in
the
final
portion
of
Corol-
lary
4.6,
(ii).
Here,
the
“R
>0
-orbits”
are
defined
relative
to
the
natural
R
>0
-
actions
on
the
Frobenioids
involved
obtained
by
multiplying
the
“arithmetic
de-
grees”
by
a
given
element
∈
R
>0
[cf.
[FrdI],
Example
6.3;
[FrdI],
Theorem
6.4,
(ii);
[IUTchI],
Remark
3.1.5].
±ell
(vi)
(Frobenius-pictures)
Let
{
n
HT
Θ
NF
}
n∈Z
be
a
collection
of
distinct
±ell
Θ
NF-Hodge
theaters
indexed
by
the
integers.
Then
by
applying
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
(iii),
we
obtain
infinite
chains
Θ
×μ
(n−1)
Θ
×μ
gau
(n−1)
.
.
.
−→
.
.
.
−→
NF
Θ
×μ
n
Θ
±ell
NF
Θ
×μ
gau
n
HT
Θ
HT
±ell
−→
−→
NF
Θ
×μ
(n+1)
Θ
±ell
NF
Θ
×μ
gau
(n+1)
HT
Θ
HT
±ell
−→
−→
NF
Θ
×μ
Θ
±ell
NF
Θ
×μ
gau
HT
Θ
HT
±ell
−→
.
.
.
−→
.
.
.
±ell
NF-Hodge
theaters.
Either
of
these
infinite
chains
of
Θ
×μ
-/Θ
×μ
gau
-linked
Θ
may
be
represented
symbolically
as
an
oriented
graph
Γ
[cf.
[AbsTopIII],
§0]
...
→
•
→
•
→
•
→
Θ
×μ
...
Θ
×μ
gau
—
i.e.,
where
the
arrows
correspond
to
either
the
“
−→
’s”
or
the
“
−→
’s”,
and
±ell
the
“•’s”
correspond
to
the
“
n
HT
Θ
NF
”.
This
oriented
graph
Γ
admits
a
natural
action
by
Z
—
i.e.,
a
translation
symmetry
—
but
it
does
not
admit
arbitrary
permutation
symmetries.
For
instance,
Γ
does
not
admit
an
automorphism
that
switches
two
adjacent
vertices,
but
leaves
the
remaining
vertices
fixed
—
cf.
the
discussion
of
[IUTchI],
Corollary
3.8;
[IUTchI],
Remark
3.8.1.
Proof.
The
various
assertions
of
Corollary
4.10
follow
immediately
from
the
defi-
nitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.10.1.
Strictly
speaking
[cf.
Remark
4.6.2,
(i)],
the
F
-prime-strips
constructed,
in
Corollary
4.10,
(ii),
from
the
theta
and
Gaussian
monoids
of
Corol-
lary
4.6,
(iv),
are
only
well-defined
up
to
an
indeterminacy,
at
the
v
∈
V
bad
,
relative
to
automorphisms
of
the
split
Frobenioid
at
such
v
∈
V
bad
that
induce
the
iden-
tity
automorphism
on
the
associated
F
×
-prime-strip.
On
the
other
hand,
such
162
SHINICHI
MOCHIZUKI
indeterminacies
may,
in
essence,
be
ignored,
since
they
are
“absorbed”
in
the
full
poly-isomorphisms
that
appear
in
the
Θ
×μ
-
and
Θ
×μ
gau
-links
of
Corollary
4.10,
(iii).
Remark
4.10.2.
(i)
Although
both
the
Θ
×μ
-
and
Θ
×μ
gau
-links
are
treated,
in
essence,
on
an
equal
footing
in
Corollary
4.10,
in
the
remainder
of
the
present
series
of
papers,
we
shall
ultimately
mainly
be
interested
in
[a
further
enhanced
version
of]
the
Θ
×μ
gau
-
×μ
link.
On
the
other
hand,
the
significance
of
the
Θ
-link
lies
in
the
fact
that
it
is
precisely
by
thinking
of
[a
further
enhanced
version
of]
the
Θ
×μ
gau
-link
as
an
object
that
is
constructed
as
the
composite
of
the
Θ
×μ
-link
with
the
operation
of
Galois
evaluation
that
one
may
establish
the
crucial
multiradiality
properties
discussed
in
[IUTchIII],
Theorem
3.11.
(ii)
At
v
∈
V
bad
,
the
Θ
×μ
-link
may
be
thought
of
as
a
sort
of
equivalence
between
the
split
theta
monoids
of
Proposition
3.1,
(i)
[cf.
also
Corollary
1.12,
(ii)]
and
certain
submonoids
of
the
constant
monoids
of
Proposition
3.1,
(ii),
equipped
with
the
splittings
that
arise
from
the
q-parameter
“q
”
[cf.
the
discussion
of
“τ
v
”
v
in
[IUTchI],
Example
3.2,
(iv)].
On
the
other
hand,
it
is
important
to
note
in
this
context
that
unlike
the
case
with
the
splittings
that
occur
in
the
case
of
the
theta
monoids,
the
splittings
that
occur
in
the
case
of
the
constant
monoids
do
not
arise
from
the
operation
of
Galois
evaluation
—
i.e.,
from
a
splitting
“H
→
G
v
”
at
the
level
of
Galois
groups
of
some
surjection
G
v
H.
In
particular,
the
splittings
in
the
case
of
the
constant
monoids
do
not
admit
a
natural
multiradial
formulation
[cf.
Remark
1.11.5;
Proposition
3.4,
(ii)],
as
in
the
case
of
the
theta
monoids
[cf.
Corollary
1.12,
(iii)],
that
allows
one
to
decouple
the
monoids
into
“purely
radial”
and
“purely
coric”
components
[cf.
discussion
of
Remarks
1.11.4,
(i);
1.12.2,
(vi)].
Remark
4.10.3.
(i)
The
“coricity
of
F
×μ
-prime-strips”
†
×μ
F
∼
→
‡
×μ
F
discussed
in
Corollary
4.10,
(iv),
amounts,
in
essence,
to
the
“coricity
of
D
-prime-
∼
strips”
†
D
→
‡
D
[cf.
Corollary
4.10,
(iv)],
together
with
the
“coricity
of
[various
quotients
by
torsion
of
]
the
units
O
×
(−)”
of
the
Frobenioids
involved
—
cf.
[IUTchI],
Corollary
3.7,
(ii),
(iii).
In
[IUTchIII],
this
coricity
of
the
units
will
play
a
central
role
when
we
apply
the
theory
of
the
log-wall
[cf.
[AbsTopIII]].
In
particular,
this
coricity
of
the
units
will
allow
us
to
compare
volumes
on
either
side
of
the
Θ
×μ
-,
Θ
×μ
gau
-links.
(ii)
Unlike
the
units
[cf.
the
discussion
of
(i)!],
the
“divisor
monoid”,
or
“value
group”,
portion
of
the
Frobenioids
involved
is
by
no
means
preserved
by
the
Θ
×μ
-,
×μ
-,
Θ
×μ
Θ
×μ
gau
-links!
Indeed,
this
“value
group”
portion
of
the
Θ
gau
-links
may
be
thought
of
as
a
sort
of
“Frobenius
morphism”
—
cf.
the
discussion
of
Remark
3.6.2,
(iii),
as
well
as
Remark
4.11.1
below.
Alternatively,
from
the
point
of
view
of
the
analogy
between
[complex
or
p-adic]
Teichmüller
theory
and
the
theory
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
163
the
present
series
of
papers,
this
portion
of
the
Θ
×μ
-,
Θ
×μ
gau
-links
may
be
thought
of
as
a
sort
of
Teichmüller
deformation
[cf.
the
discussion
of
[IUTchI],
Remark
3.9.3,
(ii)].
Indeed,
the
computation
of
the
“volume
distortion”
arising
from
this
“arithmetic
Teichmüller
deformation”
may,
in
some
sense,
be
regarded
as
the
ultimate
goal
of
the
present
series
of
papers.
(iii)
In
the
context
of
the
discussion
of
(ii),
it
is
interesting
to
note
that
if
one
restricts
the
value
group
portion
of
the
Θ
×μ
gau
-link
—
i.e.,
2
q
→
q
j
v
v
1≤j≤l
[cf.
Remark
3.6.2,
(iii)]
—
to
the
label
j
=
1,
then
the
resulting
correspondence
q
→
v
q
v
may
be
naturally
identified
with
the
“identity”
—
cf.
the
discussion
of
Remark
3.6.2,
(iii).
Put
another
way,
the
restriction
to
the
label
j
=
1
of
the
Gaussian
distribution
may
be
identified,
for
instance
at
the
level
of
realifications,
with
the
pivotal
distribution
discussed
in
[IUTchI],
Example
5.4,
(vii).
On
the
other
hand,
in
this
context,
it
is
important
to
observe
that
the
operation
of
restriction
to
various
proper
subsets
of
the
set
of
all
labels
|F
l
|
fails,
in
general,
to
be
compatible
with
the
crucial
F
±
l
-
and
F
l
-symmetries
of
Corollaries
4.5,
(iii);
4.6,
(iii);
4.7,
(ii);
4.8,
(ii)
[cf.
also
the
discussion
of
Remark
2.6.3].
n
HT
D-Θ
...
n
±ell
NF
...
|
±ell
HT
D-Θ
NF
—
(−)
—
D
n
±ell
HT
D-Θ
|
...
n
NF
...
±ell
HT
D-Θ
NF
Fig.
4.3:
Étale-picture
of
D-Θ
±ell
NF-Hodge
Theaters
Corollary
4.11.
(Étale-pictures
of
Base-Θ
±ell
NF-Hodge
Theaters)
Sup-
pose
that
we
are
in
the
situation
of
Corollary
4.10,
(vi).
(i)
Write
...
D
−→
n
±ell
HT
D-Θ
NF
D
−→
(n+1)
±ell
HT
D-Θ
NF
D
−→
...
164
SHINICHI
MOCHIZUKI
—
where
n
∈
Z
—
for
the
infinite
chain
of
D-Θ
±ell
NF-linked
D-Θ
±ell
NF-
Hodge
theaters
[cf.
Corollary
4.10,
(iv),
(vi)]
induced
by
either
of
the
infinite
chains
of
Corollary
4.10,
(vi).
Then
this
infinite
chain
induces
a
chain
of
full
poly-isomorphisms
∼
...
→
n
∼
D
→
(n+1)
∼
D
→
.
.
.
[cf.
Corollary
4.10,
(iv)].
That
is
to
say,
“
(−)
D
”
forms
a
constant
invariant
[cf.
the
discussion
of
[IUTchI],
Remark
3.8.1,
(ii)]
—
i.e.,
a
mono-analytic
core
[cf.
the
situation
discussed
in
[IUTchI],
Remark
3.9.1]
—
of
the
above
infinite
chain.
(ii)
If
we
regard
each
of
the
D-Θ
±ell
NF-Hodge
theaters
of
the
chain
of
(i)
as
a
spoke
emanating
from
the
mono-analytic
core
“
(−)
D
”
discussed
in
(i),
then
we
obtain
a
diagram
—
i.e.,
an
étale-picture
of
D-Θ
±ell
NF-Hodge
theaters
—
as
in
Fig.
4.3
above
[cf.
the
situation
discussed
in
[IUTchI],
Corollaries
4.12,
6.10].
Thus,
each
spoke
may
be
thought
of
as
a
distinct
“arithmetic
holomorphic
structure”
on
the
mono-analytic
core.
Finally,
[cf.
the
situation
discussed
in
[IUTchI],
Corollaries
4.12,
6.10]
this
diagram
satisfies
the
important
property
of
admitting
arbitrary
permutation
symmetries
among
the
spokes
[i.e.,
the
labels
n
∈
Z
of
the
D-Θ
±ell
NF-Hodge
theaters].
(iii)
The
constructions
of
(i)
and
(ii)
are
compatible,
in
the
evident
sense,
with
the
constructions
of
[IUTchI],
Corollaries
4.12,
6.10,
relative
to
the
natural
∼
identification
isomorphisms
(−)
D
→
(−)
D
>
[cf.
Corollary
4.10,
(i);
the
discussion
preceding
[IUTchI],
Example
5.4]
and
the
operation
of
passing
to
the
underlying
D-ΘNF-
[in
the
case
of
[IUTchI],
Corollary
4.12]
and
D-Θ
±ell
-Hodge
theaters
[in
the
case
of
[IUTchI],
Corollary
6.10].
Proof.
The
various
assertions
of
Corollary
4.11
follow
immediately
from
the
defi-
nitions
and
the
references
quoted
in
the
statements
of
these
assertions.
Remark
4.11.1.
The
Θ
×μ
gau
-link
of
Corollary
4.10,
(iii),
may
be
thought
of,
roughly,
as
a
sort
of
transformation
q
→
1
2
..
.
2
)
(l
q
—
cf.
the
discussion
of
Remark
3.6.2,
(iii).
From
this
point
of
view,
the
infinite
chain
of
the
Frobenius-picture
discussed
in
Corollary
4.10,
(vi),
may
be
represented
as
an
infinite
iteration
2
q
→
1
..
.
2
)
(l
q
1
2
..
.
(l
)
2
···
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
165
of
this
transformation.
By
contrast,
the
associated
étale-picture
discussed
in
Corollary
4.11
corresponds
to
a
sort
of
commutativity
involving
these
“theta
expo-
nents”
1
2
1
2
1
2
..
..
..
·
·
···
.
.
.
2
2
2
(l
)
(l
)
q
→
q
(l
)
—
cf.
the
“arbitrary
permutation
symmetries”
discussed
in
Corollary
4.11,
(ii).
In
this
context,
it
is
useful
to
recall
the
analogy
between
the
classical
Gaussian
in-
tegral
and
the
theory
of
the
present
series
of
papers
[cf.
Remark
1.12.5]
—
an
anal-
ogy
in
which
the
“order-conscious”
Frobenius-picture
corresponds
to
the
carte-
sian
coordinate
representation
of
the
Gaussian
integral,
while
the
“permutation-
symmetric”
étale-picture
corresponds
to
the
polar
coordinate
representation
of
the
Gaussian
integral.
Finally,
from
the
point
of
view
of
the
discussion
of
Remark
2
4.7.4,
the
l-torsion
that
occurs
as
the
index
set
of
the
various
“q
j
’s”
that
appear
in
the
Gaussian
monoid
of
each
Θ
±ell
NF-Hodge
theater
may
be
thought
of
as
a
sort
of
multiradial
combinatorial
representation
of
the
distinct
“arithmetic
holomorphic
structures”
corresponding
to
the
various
Θ
±ell
NF-Hodge
theaters.
Remark
4.11.2.
At
this
point,
we
pause
to
review
the
theory
developed
so
far
in
the
present
series
of
papers.
(i)
The
notion
of
a
Θ
±ell
NF-Hodge
theater
[cf.
[IUTchI],
Definition
6.13,
(i)]
is
intended
as
a
model
of
conventional
scheme-theoretic
arithmetic
geometry
—
i.e.,
more
precisely,
of
the
conventional
scheme-theoretic
arithmetic
geometry
surrounding
the
theta
function
at
primes
of
bad
reduction
∈
V
bad
of
the
elliptic
curve
over
a
number
field
under
consideration.
At
a
more
technical
level,
a
Θ
±ell
NF-
Hodge
theater
may
be
thought
of
as
an
apparatus
that
allows
one
to
construct
a
sort
of
bridge
between
the
number
field
and
theta
functions
[at
v
∈
V
bad
]
under
consideration.
From
a
more
concrete
point
of
view,
this
bridge
is
realized
by
the
Gaussian
distribution
—
i.e.,
a
globalized
version
of
the
theta
values
q
j
2
v
1≤j≤l
at
l-torsion
points
[cf.
Remark
3.6.2,
(iii)].
Here,
we
remark
that
the
term
“Gauss-
ian
distribution”
is
intended
as
an
intuitive
expression
that
includes
the
more
tech-
nical
notions
of
“Gaussian
monoids”
and
“Gaussian
Frobenioids”.
The
Gaussian
distribution
also
plays
the
crucial
role
of
allowing
the
construction
of
the
[non-
±ell
NF-Hodge
theaters
[cf.
scheme/ring-theoretic!]
Θ
×μ
gau
-link
between
distinct
Θ
Corollary
4.10,
(iii)]
—
i.e.,
between
distinct
models
of
conventional
scheme-
theoretic
arithmetic
geometry.
(ii)
Within
a
single
Θ
±ell
NF-Hodge
theater,
the
theory
of
étale
and
Frobenioid-
theoretic
theta
functions
developed
in
[EtTh]
is
applied
to
construct
a
single
con-
nected
geometric
“Kummer
theory-compatible
theater
for
evaluation
of
the
theta
function”,
whose
étale-theoretic
realization
admits
a
multiradial
formulation
[cf.
the
theory
of
§1,
especially
Corollary
1.12],
and
whose
connectedness
allows
one
to
establish
conjugate
synchronization
[cf.
the
discussion
of
Remark
2.6.1]
166
SHINICHI
MOCHIZUKI
between
the
various
copies
of
the
absolute
Galois
group
of
the
base
field
at
the
various
l-torsion
points
at
which
the
theta
function
is
evaluated.
Moreover,
this
conjugate
synchronization
satisfies
the
crucial
property
of
compatibility
with
the
F
±
l
-symmetry
[cf.
the
discussion
of
Remark
3.5.2,
as
well
as
Corollaries
4.5,
(iii);
4.6,
(iii)]
of
the
underlying
D-Θ
ell
-bridge
[cf.
[IUTchI],
Proposition
6.8,
(i)]
of
the
Θ
±ell
NF-Hodge
theater
under
consideration.
Conjugate
synchronization
plays
an
essential
role
in
establishing
the
coricity
of
the
units
[cf.
Corollary
4.10,
(iv)]
in
a
fashion
which
is
compatible
with
both
the
étale-theoretic
—
i.e.,
“an-
abelian”
—
and
abstract
monoid/Frobenioid-theoretic
—
i.e.,
“post-anabelian”
—
representations
of
the
Gaussian
monoids
[cf.
the
discussion
of
Remark
3.8.3].
Here,
we
recall
that
the
“post-anabelian”
representation
of
the
Gaussian
monoids
is
necessary
to
construct
the
Θ
×μ
gau
-link
of
Corollary
4.10,
(iii)
[cf.
Remarks
3.6.2,
(ii);
3.8.3,
(i)].
On
the
other
hand,
the
“anabelian”
representation
of
the
Gaussian
monoids
will
play
an
essential
role
when
we
apply
the
theory
of
the
log-wall
[cf.
[AbsTopIII]]
in
[IUTchIII]
[cf.
Remark
3.8.3,
(ii)].
Another
important
aspect
of
the
theory
of
Gaussian
distibutions,
at
v
∈
V
bad
,
is
the
canonical
splittings
of
the
monoids
involved
into
“unit”
and
“value
group”
components.
These
splittings
may
be
thought
of,
in
the
context
of
the
Θ
×μ
gau
-link,
as
corresponding
to
the
“non-
deformed”
[cf.
the
“coricity
of
the
units”]
and
“Teichmüller-dilated”
[cf.
the
“value
group”
portion
of
the
Gaussian
distribution]
real
dimensions
that
appear
in
classical
complex
Teichmüller
theory
[cf.
the
discussion
of
Remark
4.10.3,
(i),
(ii)].
Finally,
these
splittings
will
play
a
crucial
role
in
the
theory
of
log-shells
[cf.
[AbsTopIII]],
which
we
shall
apply
in
[IUTchIII].
(iii)
By
contrast,
the
number
fields
that
appear
in
the
underlying
ΘNF-
Hodge
theater
of
the
Θ
±ell
NF-Hodge
theater
under
consideration
[cf.
the
theory
of
[IUTchI],
§5]
will
ultimately,
in
[IUTchIII],
in
the
context
of
log-shells,
play
the
role
of
relating
—
via
the
ring
structure
of
these
number
fields
—
-line
bundles
[i.e.,
“idèlic”
or
“Frobenioid-theoretic”
line
bundles]
to
“
-line
bundles”
[i.e.,
line
bundles
thought
of
as
modules]
—
cf.
the
discussion
of
Remark
4.7.2.
Such
rela-
tionships
are
only
possible
if
one
considers
all
of
the
primes
of
the
number
fields
involved
[cf.
[AbsTopIII],
Remark
5.10.2,
(iv)].
Constructions
associated
to
these
number
fields
satisfy
the
property
of
being
compatible
with
the
F
l
-symmetry
[cf.
[IUTchI],
Proposition
4.9,
(i)]
of
the
underlying
NF-bridge
of
the
Θ
±ell
NF-Hodge
theater
under
consideration.
Unlike
the
F
±
l
-symmetry
discussed
in
(ii),
which
is
combinatorially
uniradial
in
nature
and
may
be
thought
of,
in
the
context
of
the
splittings
discussed
in
(ii),
as
being
associated
with
the
“units”,
the
F
l
-symmetry
is
combinatorially
multiradial
in
nature
and
may
be
thought
of,
in
the
context
of
the
splittings
discussed
in
(ii),
as
being
associated
with
the
“value
groups”
[cf.
the
discussion
of
Remarks
4.7.3,
4.7.4,
4.7.5].
On
the
other
hand,
[cf.
the
discussion
of
(ii)]
the
F
±
l
-symmetry
satisfies
the
crucial
property
of
being
compatible
with
conjugate
synchronization
—
a
property
which
may
only
be
established
after
one
isolates
the
prime-strips
under
consideration
from
the
conjugacy
indetermi-
nacies
inherent
in
the
global
structure
of
the
absolute
Galois
group
of
a
number
field
[cf.
Remark
4.7.2].
Put
another
way,
conjugate
synchronization
may
only
be
established
once
the
prime-strips
under
consideration
are
treated
as
objects
which
are
free
of
any
combinatorial
constraints
arising
from
the
“prime-trees”
asso-
ciated
to
a
number
field
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1].
On
the
other
hand,
one
important
property
shared
by
both
the
F
±
l
-
and
F
l
-symmetries
is
the
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
167
connectedness
of
the
global
objects
that
appear
in
the
[Θ
ell
-/NF-bridges
of]
these
symmetries.
This
connectedness
plays
an
essential
role
in
the
bookkeeping
opera-
tions
involving
the
labels
of
the
evaluation
points
[cf.
the
discussion
of
Remarks
3.5.2
and
4.5.3,
(iii),
as
well
as
[IUTchI],
Remark
4.9.2,
(i)].
In
particular,
such
bookkeeping
operations
cannot
be
implemented
if,
for
instance,
instead
of
working
with
a
global
number
field,
one
attempts
to
take
as
one’s
“global
objects”
formal
products
of
the
local
objects
at
the
various
primes
of
the
number
field
under
con-
sideration
[cf.
the
discussion
of
[AbsTopIII],
Remark
3.7.6,
(v)].
Finally,
we
recall
that
the
essential
role
played
by
these
“global
bookkeeping
operations”
gives
rise,
in
light
of
the
profinite
nature
of
the
global
geometric
étale
fundamental
groups
involved,
to
a
situation
in
which
one
must
apply
the
“complements
on
tempered
coverings”
developed
in
[IUTchI],
§2
[cf.
Remark
4.5.3,
(iii)].
(iv)
One
way
to
summarize
the
above
discussion
is
as
follows.
The
bridge
constituted
by
the
Gaussian
distribution
of
a
Θ
±ell
NF-Hodge
theater
between
theta
functions
and
number
fields
may
be
thought
of
as
being
constructed
by
dismantling
those
aspects
of
the
“characteristic
topography”
of
the
theta
functions
and
number
fields
involved
that
constitute
an
obstruction
to
relating
theta
functions
to
number
fields.
In
the
case
of
theta
functions,
the
main
obstruction
to
constructing
such
a
link
to
the
number
field
under
consideration
is
constituted
by
the
geometric
dimension
of
the
tempered
coverings
of
elliptic
curves
[at
v
∈
V
bad
]
on
which
the
theta
functions
are
defined.
This
obstruction
is
resolved
by
means
of
the
operation
of
evaluation
at
the
l-torsion
points.
Thus,
from
the
point
of
view
of
the
scheme-
theoretic
Hodge-Arakelov
theory
of
[HASurI],
[HASurII],
one
may
think
of
these
l-torsion
points
as
a
sort
of
“rough
finite
approximation”
of
the
tempered
coverings
of
elliptic
curves
under
consideration
[cf.
the
discussion
of
[HASurI],
§1.3.4].
By
contrast,
in
the
case
of
number
fields,
the
main
obstruction
to
constructing
such
a
link
to
the
theta
functions
under
consideration
is
the
“prime-trees”
arising
from
the
global
structure
of
the
number
field,
which
give
rise
to
the
conjugacy
indeterminacies
that
obstruct
the
establishment
of
conjugate
synchronization
[cf.
the
discussion
of
(iii)
above].
This
obstruction
is
resolved
by
dismantling
the
global
prime-tree
structure
of
the
number
fields
involved
by
working
with
various
prime-strips
labeled
by
elements
∈
F
l
[cf.
the
discussion
of
[IUTchI],
Remark
4.3.1].
Thus,
one
may
think
of
these
collections
of
prime-strips
labeled
by
elements
∈
F
l
as
“rough
finite
approximations”
of
the
infinite
prime-trees
associated
to
the
number
fields
involved.
At
a
more
combinatorial
level
[cf.
the
discussion
of
Remark
4.7.5],
this
dismantling
process
may
be
thought
of
as
the
process
of
dismantling
the
ring
structure
of
F
l
—
which
we
think
of
as
a
“rough
finite
approximation”
of
Z
[cf.
[IUTchI],
Remark
6.12.3,
(i)]
—
into
its
additive
and
multiplicative
components,
which
correspond,
respectively,
to
the
F
±
l
-
and
F
l
-
symmetries.
Remark
4.11.3.
In
the
context
of
the
discussion
of
Remark
4.11.2,
it
is
interesting
to
observe
that,
whereas,
from
the
point
of
view
of
the
combinatorics
of
the
F
±
l
-
and
F
l
-symmetries,
one
has
correspondences
Θ
ell
←→
,
NF
←→
—
i.e.,
the
Θ
ell
-bridge
corresponds
to
the
additive
F
±
l
-symmetry,
while
the
NF-
bridge
corresponds
to
the
multiplicative
F
l
-symmetry
—
at
the
level
of
line
bundles,
168
SHINICHI
MOCHIZUKI
one
has
correspondences
Θ
ell
←→
,
NF
←→
—
i.e.,
the
arithmetic
line
bundles
under
consideration
are
treated
multiplicatively,
via
monoids
or
Frobenioids,
in
the
context
of
the
Θ
ell
-bridge,
while
the
equivalence
of
such
-line
bundles
with
-line
bundles
may
only
be
realized
in
the
context
of
the
global
ring
structure
of
the
number
fields
associated,
via
the
theory
of
[IUTchI],
§5,
to
the
NF-bridge.
This
“juggling
of
and
”
is
reminiscent
of
the
theory
of
the
log-wall
developed
in
[AbsTopIII]
[cf.,
e.g.,
the
discussion
of
[AbsTopIII],
§I3]
and,
indeed,
may
be
thought
of
as
a
sort
of
combinatorial
counterpart
to
the
“juggling
of
and
”
that
occurs
in
the
theory
of
the
log-wall.
Remark
4.11.4.
(i)
From
the
point
of
view
of
scheme-theoretic
Hodge-Arakelov
theory,
the
l-
torsion
points
of
an
elliptic
curve
may
be
thought
of
as
a
“rough
finite
approxi-
mation”
of
the
two
real
dimensions
of
the
underlying
real
analytic
manifold
of
a
one-dimensional
complex
torus
[cf.
the
discussion
of
[HASurI],
§1.3.4].
In
scheme-
theoretic
Hodge-Arakelov
theory,
one
considers
spaces
of
functions
on
these
l-torsion
points.
The
two
dimensions
mentioned
above
then
correspond
to
a
“holomorphic
dimension”
and
a
“one-dimensional
deformation
of
this
holomorphic
dimension”
[cf.
the
discussion
of
[HASurI],
§1.4.2].
In
the
context
of
the
theory
of
the
present
series
of
papers,
we
work,
in
effect,
with
an
elliptic
curve
which
is
isogenous
to
the
given
elliptic
curve
via
an
isogeny
of
degree
l
—
i.e.,
with
“X”
as
opposed
to
“X”
—
so
that
we
may
neglect
the
“holomorphic
dimension”
mentioned
above
and
concentrate
instead
on
the
deformations
of
this
“holomorphic
dimension”
[cf.
the
discussion
of
the
Introduction
to
[EtTh]].
In
particular,
the
various
possible
values
at
the
various
l-torsion
points
at
which
the
theta
function
is
evaluated
in
the
theory
of
the
present
series
of
papers
may
be
thought
of
as
various
possible
deformations
of
the
holomorphic
structure,
while
the
specific
values
of
the
theta
function
may
be
thought
of
as
a
specific
deformation
of
the
holomorphic
structure.
Here,
we
note
that
the
parameter
“0
=
t
∈
LabCusp
±
(−)”
that
indexes
these
values
—
which,
like
the
tangent
space
to
the
original
elliptic
curve,
is
linear
which
respect
to
the
group
structure
of
the
elliptic
curve
—
descends
naturally
[especially
in
the
context
of
ΘNF-Hodge
theater!]
to
the
parameter
“j
∈
F
l
”
—
which
may
be
thought
of
as
×
2
the
“square
(F
×
)
”
of
F
,
hence,
like
the
square
of
the
tangent
space
of
the
elliptic
l
l
curve,
which
is
naturally
isomorphic
to
the
tangent
space
to
the
moduli
space
of
elliptic
curves
at
the
point
determined
by
the
elliptic
curve
in
question,
is
quadratic
in
its
dependence
on
the
linear
group
structure
of
the
elliptic
curve.
Finally,
this
point
of
view
concerning
the
values
of
the
theta
function
is
reminiscent
of
the
point
of
view
of
Remark
3.6.2,
(iii),
in
which
we
observe
that,
in
the
context
of
the
Θ
×μ
gau
-
link,
these
values
of
the
theta
function
may
be
thought
of
as
a
sort
of
“deformation
between
the
identity
and
a
Frobenius
morphism”.
The
theta
function
involved
may
then
be
thought
of
as
a
sort
of
continuous
version
[i.e.,
as
opposed
to
a
“rough
finite
approximation”]
of
such
a
deformation.
(ii)
From
the
point
of
view
of
the
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
[cf.
[AbsTopIII],
§I5],
the
portion
of
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
169
the
infinite
chain
of
Θ
×μ
-links
of
Corollary
4.10,
(vi),
parametrized
by
n
≤
0
Θ
×μ
.
.
.
−→
(n−1)
HT
Θ
±ell
NF
Θ
×μ
±ell
−→
n
HT
Θ
NF
Θ
×μ
Θ
×μ
±ell
−→
.
.
.
−→
0
HT
Θ
NF
may
be
thought
of
as
corresponding
to
the
canonical
liftings
of
p-adic
Teichmüller
theory.
That
is
to
say,
each
Θ
±ell
NF-Hodge
theater
—
which
one
may
think
of
as
representing
the
conventional
scheme
theory
surrounding
the
given
number
field
equipped
with
an
elliptic
curve
—
corresponds
to
a
hyperbolic
curve
in
posi-
tive
characteristic
equipped
with
a
nilpotent
ordinary
indigenous
bundle
[cf.
the
discussion
of
[AbsTopIII],
§I5].
The
theta
functions
that
give
rise
to
the
Θ
×μ
-links
may
be
thought
of
as
specifying
the
specific
canonical
deformation
[cf.
the
discus-
sion
of
(i)]
that
gives
rise
to
this
“canonical
lifting”.
The
canonical
Frobenius
lifting
on
this
canonical
lifting
may
be
thought
of
as
corresponding
to
the
theory
to
be
developed
in
[IUTchIII].
From
this
point
of
view,
the
passage
theta
functions,
number
fields
Gaussian
distributions
[cf.
the
discussion
of
Remark
4.11.2]
effected
in
the
theory
of
the
present
series
of
papers
presented
thus
far
—
i.e.,
at
a
more
concrete
level,
the
passage,
via
Hodge-Arakelov-theoretic
evaluation,
from
the
above
semi-infinite
chain
to
the
corresponding
semi-infinite
chain
Θ
×μ
gau
.
.
.
−→
(n−1)
±ell
HT
Θ
NF
Θ
×μ
gau
−→
n
±ell
HT
Θ
NF
Θ
×μ
gau
Θ
×μ
gau
−→
.
.
.
−→
0
HT
Θ
±ell
NF
of
Θ
×μ
gau
-links
—
may
be
thought
of
as
corresponding
to
the
passage
MF
∇
-objects
Galois
representations
in
the
case
of
the
canonical
indigenous
bundles
that
occur
in
p-adic
Teichmüller
theory
—
cf.
the
discussion
of
[pTeich],
Introduction,
§1.3,
§1.7;
the
discussion
in
[HASurI],
§1.3,
§1.4,
of
the
relationship
between
such
canonical
indigenous
bundles
in
the
case
of
the
moduli
stack
of
elliptic
curves
and
the
scheme-theoretic
Hodge-
Arakelov
theory
of
[HASurI],
[HASurII].
Put
another
way,
it
corresponds
to
the
passage
from
thinking
of
the
“canonical
lifting”
as
a
curve
equipped
with
the
MF
∇
-
object
constituted
by
a
canonical
Frobenius-invariant
indigenous
bundle
to
thinking
of
the
“canonical
lifting”
as
a
curve
equipped
with
a
canonical
Galois
representation,
i.e.,
a
canonical
crystalline
representation
[that
is
to
say,
a
representation
that
happens
to
arise
from
an
MF
∇
-object]
of
the
arithmetic
fundamental
group
of
the
generic
fiber
of
the
curve
into
P
GL
2
(Z
p
).
(iii)
The
analogy
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory
may
also
be
seen,
at
a
more
technical
level,
in
the
following
correspondences
between
various
aspects
of
the
theory
presented
thus
far
in
the
present
series
of
papers
and
various
aspects
of
the
theory
of
[CanLift],
§3
[cf.
also
Remark
4.11.5
below]:
(a)
The
discussion
of
(ii)
above
is
reminiscent
of
the
important
role
played
by
the
canonical
Galois
representation
in
the
absolute
p-adic
anabelian
theory
of
[CanLift],
§3
[cf.
the
proof
of
[CanLift],
Lemma
3.5].
170
SHINICHI
MOCHIZUKI
(b)
In
light
of
the
important
role
played,
in
the
present
series
of
papers,
by
mono-theta-theoretic
cyclotomic
rigidity
[which
was
reviewed
in
Definition
1.1,
(ii);
Remark
1.1.1],
it
is
perhaps
of
interest
to
recall
[cf.
Remark
1.11.6]
the
important
role
played
by
cyclotomic
rigidity
isomor-
phisms
in
the
theory
of
[CanLift],
§3,
via
the
theory
of
[AbsAnab],
§2
[cf.,
especially,
[AbsAnab],
Lemmas
2.5,
2.6].
On
the
other
hand,
at
the
level
of
direct
correspondences
between
the
theory
of
the
present
series
of
papers
and
p-adic
Teichmüller
theory,
it
is
perhaps
better
to
think
of
mono-theta-theoretic
cyclotomic
rigidity
as
corresponding
to
the
local
uniformizations
arising
from
the
canonical
indigenous
bundle
[cf.
the
discussion
of
Remark
3.6.5,
(iii)].
(c)
The
important
role
played,
in
the
present
series
of
papers,
by
the
“two-
dimensional
symmetry”
constituted
by
the
F
±
l
-
and
F
l
-symmetries
—
whose
two-dimensionality
may
be
thought
of
as
corresponding
to
the
two
real
dimensions
of
the
complex
upper
half-plane
[cf.
the
discussion
of
[IUTchI],
Remark
6.12.3,
(iii)]
—
is
reminiscent
of
the
important
role
played
in
the
theory
of
[CanLift],
§3,
in
effect,
by
the
vanishing
of
the
zero-th
group
cohomology
module
H
0
(Ad(−))
of
the
canonical
Galois
representation
associated
to
the
canonical
indige-
nous
bundle
—
cf.
the
various
geometric
conditions
over
the
ordinary
locus
and
at
the
supersingular
points
of
the
mod
p
representations
con-
sidered
in
[CanLift],
Lemma
3.2.
That
is
to
say,
the
F
±
l
-symmetry
may
be
regarded
as
corresponding
to
the
unipotent
monodromy
over
the
ordinary
locus
1
∗
0
1
∼
→
F
p
—
which
is
isomorphic
to
the
additive
group
underlying
F
p
—
while
the
F
l
-symmetry
may
be
regarded
as
corresponding
to
the
toral
mon-
odromy
at
the
supersingular
points
∗
0
−1
0
∗
∼
→
F
×
p
—
which
is
isomorphic
to
the
multiplicative
group
F
×
p
and
arises
from
ex-
tracting
a
(p
−
1)-th
root
of
the
Hasse
invariant.
Moreover,
the
“intuitive,
conventional”
nature
of
the
theory
over
any
single
connected
component
of
the
ordinary
locus
—
a
theory
which
allows
one,
for
instance,
to
con-
struct
q-parameters
—
is
reminiscent
of
the
uniradial
nature
of
the
F
±
l
-
symmetry,
while
the
fact
that
supersingular
points
lie
simultaneously
on
irreducible
components
obtained
as
closures
of
distinct
connected
com-
ponents
of
the
ordinary
locus
is
reminiscent
of
the
multiradiality
—
i.e.,
compatibility
with
simultaneous
execution
in
distinct
Hodge
theaters
—
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
171
of
the
F
l
-symmetry
[cf.
the
discussion
of
Remark
4.7.4].
The
above
dis-
cussion
is
summarized,
at
the
level
of
keywords,
in
Fig.
4.4
below.
F
±
l
-symmetry
F
l
-symmetry
additive
multiplicative
uniradial
multiradial
monodromy
over
the
ordinary
locus
monodromy
at
the
supersingular
points
Fig.
4.4:
Correspondence
of
symmetries
with
p-adic
Teichmüller
theory
(d)
The
important
role
played,
in
the
present
series
of
papers,
by
conjugate
synchronization
at
the
various
evaluation
points
of
the
theta
function
—
which
gives
rise,
in
the
form
of
the
Gaussian
distribution,
to
the
links
between
the
various
Θ
±ell
NF-Hodge
theaters
in
the
second
semi-
infinite
chain
that
appeared
in
the
discussion
of
(ii)
—
is
reminiscent
of
the
important
role
played
in
the
theory
of
[CanLift],
§3,
by
the
description
given
in
[CanLift],
Lemma
3.4,
of
the
first
group
cohomology
module
H
1
(Ad(−))
of
the
canonical
Galois
representation
associated
to
the
canonical
indige-
nous
bundle
—
whose
“slope
−1
portion”
may
be
thought
of
as
governing
the
“links”
between
the
“mod
p
n
”
and
“mod
p
n+1
”
portions
of
the
canon-
ical
Galois
representation,
as
it
is
considered
in
the
proof
of
[CanLift],
Lemma
3.5.
Here,
we
note
that
this
description
may
be
summarized,
in
effect,
as
asserting
that
the
slope
−1
portion
in
question
is,
up
to
tensor
product
with
an
unramified
Galois
representation,
isomorphic
to
a
direct
product
of
3g
−
3
+
r
copies
of
F
p
(−1)
[where
the
“(−1)”
denotes
a
Tate
twist]
—
a
situation
that
is
reminiscent
of
the
l
synchronized
copies
of
cyclotomes
that
occur
in
the
context
of
the
evaluation
of
the
theta
function
considered
in
the
present
series
of
papers.
Moreover,
the
deformations
of
the
canonical
Galois
representation
parametrized
by
this
module
“H
1
(Ad(−))”
may
be
thought
of
as
corresponding,
in
the
theory
of
the
present
series
of
papers,
to
the
“independent
Aut(G
v
)-indeterminacies”
[i.e.,
for
v
∈
V
non
]
that
occur
at
each
label
∈
F
l
when
one
consider
multi-
radial
representations
of
Gaussian
monoids
—
cf.
the
theory
of
[IUTchIII],
§3;
[IUTchIII],
Remark
3.12.4,
(iii).
[Here,
we
note
that,
in
fact,
the
various
“−1’s”
in
(d)
should
be
replaced
by
“1’s”
—
cf.
Remark
4.11.5
below.]
Finally,
we
observe,
with
regard
to
(d),
that
the
172
SHINICHI
MOCHIZUKI
description
in
question
that
appears
in
[CanLift],
Lemma
3.4,
may
be
thought
of
as
a
reflection
of
the
ordinarity
[i.e.,
as
opposed
to
just
admissibility]
of
the
positive
characteristic
nilpotent
indigenous
bundle
under
consideration,
hence
is
reminiscent
of
the
discussion
of
[AbsTopIII],
Remark
5.10.3,
(ii),
of
the
correspondence
between
ordinarity
in
p-adic
Teichmüller
theory
and
the
theory
of
the
étale
theta
function
developed
in
[EtTh].
Remark
4.11.5.
We
take
this
opportunity
to
correct
a
few
notational
errors
in
the
statement
of
the
condition
(†
M
)
of
[CanLift],
Lemma
3.4,
which,
however,
do
not
affect
the
proof
of
this
lemma
in
any
substantive
way.
The
subquotient
“G
2
(M
)”
(respectively,
“G
−1
”)
should
have
been
denoted
“G
−2
(M
)”
(respectively,
“G
1
”).
The
subquotient
G
−2
(M
)
(respectively,
G
1
)
is
isomorphic
to
the
tensor
product
of
an
unramified
module
with
a
Tate
twist
F
p
(−2)
(respectively,
F
p
(1)).
That
is
to
say,
there
is
a
sign
error
in
the
Tate
twists
stated
in
(†
M
).
Finally,
in
order
to
obtain
the
desired
dimensions
over
F
p
,
one
must
replace
the
cohomology
module
def
“M
=
H
1
(Δ
X
log
,
Ad(V
F
p
))”
by
the
submodule
of
this
module
consisting
of
elements
whose
restriction
to
each
of
the
cuspidal
inertia
groups
of
Δ
X
log
is
upper
triangular
with
respect
to
the
filtration
determined
by
the
nilpotent
monodromy
action
on
V
F
p
[i.e.,
by
the
cuspidal
inertia
group
in
question].
That
is
to
say,
an
elementary
computation
shows
that
the
operation
of
restriction
to
this
submodule
has
the
effect
of
lowering
the
dimension
of
G
−2
(M
)
from
3g
−
3
+
2r
to
3g
−
3
+
r,
as
desired.
INTER-UNIVERSAL
TEICHMÜLLER
THEORY
II
173
Bibliography
[Lehto]
O.
Lehto,
Univalent
Functions
and
Teichmüller
Spaces,
Graduate
Texts
in
Mathematics
109,
Springer-Verlag
(1987).
[pOrd]
S.
Mochizuki,
A
Theory
of
Ordinary
p-adic
Curves,
Publ.
Res.
Inst.
Math.
Sci.
32
(1996),
pp.
957-1151.
[pTeich]
S.
Mochizuki,
Foundations
of
p-adic
Teichmüller
Theory,
AMS/IP
Studies
in
Advanced
Mathematics
11,
American
Mathematical
Society/International
Press
(1999).
[pGC]
S.
Mochizuki,
The
Local
Pro-p
Anabelian
Geometry
of
Curves,
Invent.
Math.
138
(1999),
pp.
319-423.
[HASurI]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
I,
Arithmetic
Fundamental
Groups
and
Noncommutative
Algebra,
Proceedings
of
Symposia
in
Pure
Mathematics
70,
American
Mathematical
Society
(2002),
pp.
533-569.
[HASurII]
S.
Mochizuki,
A
Survey
of
the
Hodge-Arakelov
Theory
of
Elliptic
Curves
II,
Algebraic
Geometry
2000,
Azumino,
Adv.
Stud.
Pure
Math.
36,
Math.
Soc.
Japan
(2002),
pp.
81-114.
[AbsAnab]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Hyperbolic
Curves,
Galois
Theory
and
Modular
Forms,
Kluwer
Academic
Publishers
(2004),
pp.
77-122.
[CanLift]
S.
Mochizuki,
The
Absolute
Anabelian
Geometry
of
Canonical
Curves,
Kazuya
Kato’s
fiftieth
birthday,
Doc.
Math.
2003,
Extra
Vol.,
pp.
609-640.
[AbsSect]
S.
Mochizuki,
Galois
Sections
in
Absolute
Anabelian
Geometry,
Nagoya
Math.
J.
179
(2005),
pp.
17-45.
[SemiAnbd]
S.
Mochizuki,
Semi-graphs
of
Anabelioids,
Publ.
Res.
Inst.
Math.
Sci.
42
(2006),
pp.
221-322.
[CombGC]
S.
Mochizuki,
A
combinatorial
version
of
the
Grothendieck
conjecture,
Tohoku
Math.
J.
59
(2007),
pp.
455-479.
[Cusp]
S.
Mochizuki,
Absolute
anabelian
cuspidalizations
of
proper
hyperbolic
curves,
J.
Math.
Kyoto
Univ.
47
(2007),
pp.
451-539.
[FrdI]
S.
Mochizuki,
The
Geometry
of
Frobenioids
I:
The
General
Theory,
Kyushu
J.
Math.
62
(2008),
pp.
293-400.
[FrdII]
S.
Mochizuki,
The
Geometry
of
Frobenioids
II:
Poly-Frobenioids,
Kyushu
J.
Math.
62
(2008),
pp.
401-460.
[EtTh]
S.
Mochizuki,
The
Étale
Theta
Function
and
its
Frobenioid-theoretic
Manifes-
tations,
Publ.
Res.
Inst.
Math.
Sci.
45
(2009),
pp.
227-349.
[AbsTopI]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
I:
Generalities,
J.
Math.
Sci.
Univ.
Tokyo
19
(2012),
pp.
139-242.
[AbsTopII]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
II:
Decomposition
Groups
and
Endomorphisms,
J.
Math.
Sci.
Univ.
Tokyo
20
(2013),
pp.
171-269.
174
SHINICHI
MOCHIZUKI
[AbsTopIII]
S.
Mochizuki,
Topics
in
Absolute
Anabelian
Geometry
III:
Global
Reconstruc-
tion
Algorithms,
J.
Math.
Sci.
Univ.
Tokyo
22
(2015),
pp.
939-1156.
[IUTchI]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
I:
Construction
of
Hodge
Theaters,
RIMS
Preprint
1756
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
[IUTchIII]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
III:
Canonical
Splittings
of
the
Log-theta-lattice,
RIMS
Preprint
1758
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
[IUTchIV]
S.
Mochizuki,
Inter-universal
Teichmüller
Theory
IV:
Log-volume
Computa-
tions
and
Set-theoretic
Foundations,
RIMS
Preprint
1759
(August
2012),
to
appear
in
Publ.
Res.
Inst.
Math.
Sci.
[NSW]
J.
Neukirch,
A.
Schmidt,
K.
Wingberg,
Cohomology
of
number
fields,
Grundlehren
der
Mathematischen
Wissenschaften
323,
Springer-Verlag
(2000).
[Szp]
L.
Szpiro,
Degrés,
intersections,
hauteurs
in
Astérisque
127
(1985),
pp.
11-28.
Updated
versions
of
preprints
are
available
at
the
following
webpage:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html